Absorption and refraction in germanium at 293°K in the range 12–50 cm−1

ABSORPTION AND REFRACTION AT 293°K IN THE RANGE

IN GERMANIUM 12-50 cm-’

J. R. BIRCHand C. C. BRADLEY Division of Electrical

Science. National

Physical Laboratory.

Teddington.

Middlesex.

and M. F. KIMMITT Department

of Physics University

(Recked

of Essex. (‘okhester.

30 April

Essex, England

1974)

Abstract-Submillimetre wavelength transmission and reflection measurements on specimens of both II- and p-type germanium are presented for a range of resistivitics from 0055 to a30 Qm at 293°K. The absorption and refraction spectra are deduced from these measurements and a comparison with free charge carrier theory shows reasonable agreement. especially at the longer wavelengths.

INTRODUCTION

The propertics of germanium at submillimetre wavelengths are of interest in a number of ways. Due to the availability of high purity material having low power absorption it is often used as a window material. although reflection losses are high unless a single frequency Ctalon is constructed. A number of authors have measured the absorption and reflectivity of relatively pure germanium at 300 and 4.2 K. ‘lo’ The present work arose from a different application where it was found necessary to know the optical constants over a wider range of purity for both n- and p- type material at room temperature and this information was not available. Bradley and Kimmitt (to be published) have shown that it is possible to observe room temperature photon drag detection in germanium with submillimetre wavelength radiation from C.W.HCN (337 Itm) and H,O (28 itm) lasers, although such experiments have hitherto been limited to the 10 jlrn region’“.4’ and the 3.5 to 5.5 /drn region (Shack. private communication). The interest in this detector lies mainly in its rcsponse time (~0.5 nscc at 10.6 ilm’“)and room temperature operation since its NEP (10m3 WHz- L at IO.6 jtm. 10 ’ WHz- ; at 337 /lrn for a I x 1 mm’ area detector) is poor in comparison with other detectors. The operation of the detector is based upon momentum transfer from photons to free charge carriers and for the purposes of interpolating the data obtained it is necessary to know the appropriate absorption coefficient and refractive index. MEASUREMENTS

The single crystals of germanium used in these experiments were prepared in the usual manner by pulling from a high purity melt. The impurities were gallium (p-type) and antimony (!I-type). Resistivity measurements on disc-shaped specimens were made by a four probe technique and indicated a very high degree of homogeneity in the crystals. 1x9

I90

J. R. BIRCH, C. C. BRADLEY and M. F. KIMMITT Table

1. Physical

Specimen (Type)

and electrical

parameters

Thickness (mm)

Resistivity (Q.m)

2.135 3.36 SGO 10.12 1.33 3.20 5.80 1.oo * Experimental

of the specimens

error on all mobility

0060 0060 0.195 0,190 0.055 0.25 023 0.30

at 293°K. Mobility* (m’.V-‘.sec-‘) 0.18 0.18

k 0;003 +_ 0003 2 0.005 * 0.005 * 0.003 + 0.01 + 0.01 * 0.01

0.19 0.19 0.38 0.38 0.38 0.38

values is + 0.005.

Power transmission measurements were made on four n-type and four p-type single crystal germanium specimens. Each was roughly disc-shaped with plane-parallel end surfaces. The relevant pHysica and electrical parameters are given in Table 1. The measurements were made at 293°K in the spectral range between 12 and 50 cm- ’ using an NPL niodular Michelson interferometer(h) and a Putley-type indium antimonide photoconductive detector. (7)Phase modulation@’ of the radiation passing through the interferometer was used and the entire optical path evacuated to remove the strong absorptionsassociated with the pure rotational spectrum of atmospheric water vapour. The specimens were introduced into the exit beam of the interferometer at the detector as required and the transmission data are presented in Fig. l(a)‘and (b) below. The continuous increase in transmission to higher wavenumbers is consistent with absorption dominated by free charge carrier effects. Some of the curves exhibit small channelled-spectrum effects with a fringe spacing consistent with multiple-beam interference in the window of the detector cryostat. In order to calculate the absorption coefficient from the transmission data it is necessary to distinguish between the reflective and absorptive contributions to the transmission

3.20 mm. 0.25 Rm

I.33 mm, 0.055 Rm

580 mm, 0.23 Rm

oL__L_ 10

20

40 30 Wavenumber [cm-‘1 Fig. l(a)

50

Absorption

and refraction

Wavrnumkr

in germanium

(cm-tl

Fig. I(b) Fig. I. power transmission spectra at 293 K of the germanium specimens. (a) P-type (b) P-type. Spectral resolution is 1 cm-’ for all specimens except the 1GOmm. 0.30Qm n-type for which it is 2 cm- ‘. Note the break in the ordinate axis in Fig. l(a) between power transmissions of D2 and @3. The up@ portion from 0.3 to 0.4 refers to the 100 mm, 0.30 Rm specimen only, the lower portion to the other three.

losses. This may be done with the help of the power reflection spectrum of each specimen and the technique is described below. The reflection measurements were made at 293°K and normal incidence on a Michelson interferometer which had been adapted for such t~se,‘~’and the radiation again detected with a Putley-type photoconductor.

080

t

t r .Z .?

78 ; L

060

040 _

0.20 -

Fig. 2. Normal incidence power reflectivity spectra at 293’K of the IGO mm, @30 Rm n-type specimen ( x ) and the 3.36 mm 006 Rm p-type specimen ( + ). The continuous curves are the lamella reflectivity calculated from free charge carrier theory as described in the text.

The reflectivity spectra of one n-type and one p-type specimen are shown in Fig. 2. The p-type specimen was small and as a result of the low signal-to-noise ratio in the recorded interferograms data are only presented from 27-48 cm- ’ compared to the 9-48 cm- ‘ span for the larger n-type specimen. The continuous curves represent the lamella reflectivity calculated from a model of the complex relative permittivity using the measured electrical parameters. The form of the model used will be described in the next section.

J. R. BIRCH. C. C. BHAIX.~.Yand M. F. KIMWIT

192 THEORY

OF

FREE

CHARGE

CARRIER

ABSORPTION

The classical model of free charge carrier absorption is derived from the complex relativc permittivity of free charge carriers in ;I medium of real relative permittivity cf.. This is given by q(Q) = E’ -

j&’

=

1 1 -

E,,

7

_

2°F -

to((!j -

(1)

I\‘) i

as a function of angular frequency (‘1.In this expression 1’is the free carrier scattering frcquency and w,, the shielded plasma frequency which is given in terms of the free carrier number density A’and effective mass III* by

where e is the electronic charge and Emthe pcrmittivity of free space. The scattering frequency appearing in equation (3) is related to the free carrier scattering time r by \‘5 = 1 (3) The form of equation (1) assumes r to be an energy independent quantity that is the same for both d.c. and high frequency measurements. (This may not be exactly true but the agreement between experiment and theory is an indication that it approximately holds.) Equation (1) can also be expressed in terms of the refractive and absorption indices II and k.

The following well-known relationships for mobility b resistivity p, power absorption coefficient r and wavenumber CJmay be used to re-arrange equations (lH3) into an expression for free carrier power absorption in terms of the electrical properties of the semiconductor cc= eT IN*

(4) ci. = 4nok (!) (T

=

--

‘nc

(with c the speed of light i,t wcc~o.) Thus. 1 z = _ C2_ . c2- 7 &&r!&pT,,rr;l; pt1tr, (The absorption spectrum represented by equation wavenumber and in the limit e’ < 47r’c2a2p2(,n*)’

(5)

r‘- .

(5) decreases

with

increasing (6)

we obtain the well-known wavelength-squared variation of 2). The refractive index, n. appearing in equation (5) is also a function of the electrical parameters p and k but has not here been given in its full functional form as this is cumbersome. Howcvcr. in the calculations of model values for x the full functional form of II is used.

Absorption and refraction in germanium CALCULATIONS

OF

THE

OPTICAL

193

CONSTANTS

The power transmission, TL, of a weakly absorbing lamella of thickness b is given by(‘O) T = L

(1 - R)2 ezb

_

R2

e-xb

(7)

when channel spectra are not resolved, where R is the interface power reflectivity at normal incidence and given by R = (n - 1)2 + kZ

(n + 1)2 + k2 in terms of the optical constants. If both R and TL are measured then c( can be extracted from z* quite readily. In these measurements however, we have measured the lamella reflectivity at normal incidence, R,,, which is given by R = R(l + eCzzb) - 2R2 e-2abcos 24, L

1

_

R2,-2ab

for unresolved channel spectra. (I’) In this & is the phase change on reflection at a single vacuum-medium interface and is given by tan 4, =

2k

!I2 + k” - 1

(10)

In general it is not possible to construct R from measured R,.-valuesin an exact manner. The measurements reported herein are of TL and RI_ and in order to derive 51values from the measured T,-values the following procedure was adopted to determine R. Using only the measured electrical parameters mobility and resistivity a free charge carrier model calculation of R,, was performed for the two specimens whose rellectivities were measured. These are represented by the continuous curves of Fig. 2 and follow closely the experimental points. As a result of this excellent agreement betweenthe experimental and the calculated lamella reflectivities we shall assume that the interface reflectivity, R, of these specimens is,also that which can be caldulated from the free charge carrier model using only the measured mobility and resistivity and shall then use these calculated values to derive the ‘experimental’ absorption coefficient from the lamella transmission. The two specimens whose reflectivities were measured have plasma resonances lying in the microwave region and there is no significant dispersion due to these in the submillimetre wavelength region. The observed variation in lamella reflectivity arises from the spectral variation of the absorption suffered by the multiple reflected rays within the lamella. The values of the refractive index thus calculated from the model which go to make up the assumed interface reffectivities are presented in Table 2 and the values for the t?-type specimen agree well with those of Loewenstein”’ measured by a channel-spectrum technique. The plasma resonance is the main dispersive mechanism in germanium at these wavelengths and as we see from Fig. 2 that for these resistivities it falls in the microwave region we neglect resistivity differences and assume that the refractive indices calculated for the 0.30 Szm ntype specimen apply to all the n-type specimens and likewise for the 0.06 am p-type specimen. Using these calculated interface reflectivities the ‘experimental’ absorption coefficients of the specimens have been derived from the measured lamella transmissions according

J. R. BIRCH,C. C. BRADLEY and M. F. KIMMITT

194

Tabte 2. Calculated refractive index variation for a @30Qm n-type and a 0.06 ilm p-type specimen. Refractive index 0.06 Rm 030 Qm

Wavenumber (cm- ‘1

Wwe)

(n-type)

3.944 3.969 3.982 3.989 3,992

3.990 3.993 3.996 3,997 3.998

10 20 30 40 50

to equation (7) and are presented in Fig. 3. The continuous curves of Fig. 3 represent the values of LXfor these specimens calculated solely from the free charge carrier model using only the measured resistivities and mobilities. All of the model calculations were made assuming a value of 16.0 for EL,,@12 for the n-type effective mass ratio and 032 for the p-type effective mass ratio. DISCUSSION

The experimental and calculated curves of Fig. 3 show fair agreement, especially below At higher wavenumbers there is a consistent tendency for the experimental

25 cm-‘.

” - :yp*

580 mm 0.23 ilm 3.2 mm 0.25 fim 1.0mm 0.3Onm 1

I

30 Wavenumber

10 km”1

Fig. 3(a)

I 50

Absorption

and refraction

in germanium

2.135 3 36 5.00 10.12

I 30 Wavrnumber [cm-‘)

I LO

mm mm mm mm

0.06 nm 0.06Ilm 0~19snm 049 nm

1 50

Fig. 3(b) Fig. 3. Power absorption spectrum at 293’K for the germanium specimens calculated from the measured transmission data. (a) tt-type (b) p-type. The continuous curves represent the power absorption coefficient calculated from free charge carrier theory.

curves to be greater than the calculated curves. This disagreement is not unexpected in view of the simple model assumed which does not, for instance, take account of light hole contributions in p-type material or allow an energy dependence to the relaxatibn time t. The lamella reflectivities of Fig. 2, on the other hand agree excellently with the predictions based on the electrical parameters of the specimens which again is not surprising as the lamella reflectivity is not as sensitive as the lamella transmission to variations in the absorption coefficient and hence also not as sensitive to departures from the free charge carrier model. One possible cause for this discrepancy between the measured and calculated absorption coefficients that was investigated was the cfrect of systematic errors. To check on this the tr~~nsmission rneasurcillcnt~ on one specimen (I mm. @3052 m tt-type) were made on an entirely different apparatus. This used a Rollin-type indium antimonide photoconductar”” having a different dewar and light pipe optics to the original Putley detector, and an interferometer of the polarisation type described by Martin and Puplett.“3’ These mcasurcments give the results shown in Figs. 1 and 3 for this specimen and in Fig. 3 we still see the same qualitative form of bchaviour exhibited by the other specimens of increasing disagreement at higher wavenumbers. In the measured absorption values there is also a disparity between the values obtained for two specimens of equal resistivity but different thicknesses. The individual curves are the averages of at least six runs and the uncer~jnty in each point is less than + 5 per cent. As the possible errors in resistivity (< f O-01Qm) will only account for a small amount

196

J. R. BIRCH, C. C. BRADLEY and M. F. KIMMITT

Absorption

and refraction

in germanium

197

of this difference it is not readily explained. A possible explanation is that this is due to differing surface damage effects changing the surface reflectivities. It can be shown, for instance, that a difference of 094 in power reflectivities which have been assumed identical will result in a 013 fractional error in power absorption. This could obviously be a contributory factor to the disparity, although at these wavelengths greater than IOOpm one would expect to see the surface damage. Finally, as an ai’d to experimentalists and for clarity we present the absorption data of Fig. 3 in Fig. 4 in the form ofabsorption coefficient as a function of conductivity for several wavenumbers. The data points for identical resistivities have been averaged as have those for the 0.23 and 0.25 Rm n-type specimens. The straight lines merely join up the points and where these extrapolate back to the origin this has been indicated by a dashed line. Although only four of these lines actually do this all the lines tend to the origin thereby illustrating reasonable agreement with the inverse resistivity proportionality predicted by equation (5) for constant wavenumber. .4ck,1ow/~~!ye,t1r,1r~-We wish to thank Mr. M. N. Afsar of Birkbeck College, London for the transmission measurement on the 030R m n-type specimen. and Dr. J. Chamberlain of NPL for his helpful comments on the text.

REFERENCES 1. ZWERDLING,S. and J. P. THERIAULT,Infrared Phys. 12. 165 (1972). 2. LOEWENSTEIN, E. V., D. R. SMITH and R. L. MORGAN.Appl. Opr. 12,398 (1973). 3. GIBBON,A. F.. M. F. KIMMITT and A. C. WALKER,Appl. Phys. Lefts 17, 75 (1970). 4. DANISHEVSKII. A. M.. A. A. KASTALSKII,S. M. RYVKIN and I. D. VARASHETSKII, Sue. Ph?;s. JE:TP. ( 1970). 5. GIIISON. A. F.. M. F. KIMMITT and B. NOHRIS. Appl. Ph.vs. Lctrs (in press). 6. GANTRY. G. W., H. M. EVANS,J. CHAMBERLAIN and H. A. GEBBIE, fnfiured Ph.w. 9, 85 (1969). 7. Pr:rrru. E. H.. J. Phys. Chcm. Solids. 22. 241 ( 1961). X. CIIAMI)I:KLAIN. J.. In/bed Phys. 11, 25 (1971). 9, BIRCII. J. R.. lnfrured PhJfs. 12, 29 (1972). 10. Moss. T. S.. Opficul Proprrries oJSrmiconductor.s p. 14. Butterworths London (1969). I I, Bt
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