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High-resolution measurement of the germanium absorption edge under pressure
Physica 60 (1972) 235-243
0 North-Holland Publishing Co.
HIGH-RESOLUTION
MEASUREMENT
ABSORPTION
EDGE UNDER
N. J. TRAPPENIERS
OF THE GERMANIUM PRESSURE*
and R. VETTER’
Van der Waals-laboratorium, Universiteit van Amsterdam, Nederland (208th publication of the Van der Waals Fund)
Received 15 October 1971
Synopsis The pressure coefficient of the fundamental energy gap of germanium, aE&p, has been obtained with a high-resolution grating spectrometer. The value found, when the different phonon components are taken into account, is (5.39 + 0.1) x 1OWjeV/bar. The analysis also yields a value of the pressure coefficient of the direct gap at k = 0: aE,,/ap = (13.1 f 2) x 1O-6 eV/bar.
1. Introduction. The pressure shift of the absorption edge of germanium near 0.67 eV has been investigated by a number of authors. In a previous publicationl) we have given a survey of the results of these investigations from which it was obvious that there is a considerable variation in the data. It was also shown there that from optical measurements a reliable value of the pressure coefficient of the fundamental energy gap &??d/+ can be obtained. This result is confirmed by the present investigation, in which the prism spectrometer of the former experiment has been replaced by a high-resolution grating spectrometer. It leads to the conclusion that there is no discrepancy between the optical and electrical determinations of &Yo/+. In the present work again most of the attention was paid to the low-absorptivity tail of the edge in order to reduce the errors of extrapolation towards the value a = 0 of the absorptivity. In this region a fine structure is known to exi&) due to phonon-assisted indirect transitions from the valencband top to the minimum of the conduction band. This structure was found’) to persist up to the highest pressures of the experiments (1000 bar), but the details are well resolved only with the high-resolution spectrometer. The shape change under pressure of the absorptivity curves below 1 cm-l, which is due to the pressure dependence of the intermediate states of the indirect transitions?), is clearly detected. An attempt has been made to analyse the curves into the different phonon+ More details on this subject may be found in ref. 12. * Present address: Laboratorium voor Metaalkunde, Technische Hogeschool 235
Delft.
236
N. J. TRAPPENIERS
AND
R. VETTER
assisted components, which is important for a proper interpretation of the pressure shift in terms of aE,/ap. Moreover, from the shape change a value of the pressure coefficient can be found for E,,, the next-higher valency-to conduction-band transition. It can be shown from the theory of indirect transitions in germanium that the absorptivity, a, may be expressed asz)
where w denotes the frequency of the infrared radiative flux and i runs over the 4 phonon components both in annihilation and creation; each Ci is a constant containing density of states effective masses, refractive index, and mass density of germanium and the energies of the respective phonons; iW:b denotes the combined dipole and lattice scattering elements; fioi is the energy difference to the intermediate state;fi is a basic shape function, which is assumed to contain no other pressure-dependent parameters than E,i; Eti is the threshold energy of the respective phonon components, containing the phonon energy Ephef and the exciton binding energy Eex,i in the expression
gi is defined by the last identity of (l), and it is seen to constitute a pressuredependent multiplier. Most of the factors in this multiplier have a relative pressure coefficient of the order of lob6 per bar. The pressure dependence of the matrix elements is unknown. However, in the case of the longitudinal acoustic (LA) phonon, all these pressure coefficients are assumed to be small compared with that of (kiwi - fi~)~. The LA annihilation transition is known to proceed through the rzS conduction-band minimum (see, e.g., ref.4), which lies 0.805 eV above the valency-band maximum at rz5, (ref.2), and has a pressure coefficient aE,/ap of (13 f 1) x 10m6 eV/bar (ref. 5). This will change the multiplier by about 100 x 10S6 per bar. The transverse optical (TO) phonon component proceeds through the LJP intermediate state4). This is separated from the L1 conduction-band minimum by 2.1 eV, which is the major factor in reducing the effect of the multiplier again to 10m6 per bar in the TO case. The longitudinal optical (LO) phonon component is expected to be forbidden”), but from an argument similar to the one given by Lax and Hopfield6) it should become allowed through a A, intermediate state when the transition moves away from the threshold. In that case the LO component also has a nearly pressure-independent multiplier. The absorptivity curves below 1 cm-l consist of the TO, LO, and LA annihilation components, all of which will be submitted to the same shift of the threshold under pressure, as E,,h,i and Eex,* in (2) are found to have negligible pressure coefficients; however, only the first two components will shift purely
GERMANIUM
ABSORPTION
EDGE UNDER
PRESSURE
231
horizontally and the LA annihilation component will have a multiplier that decreases with pressure, thus complicating the measurement of a&/~$. 2. Experimental. The germanium absorptivity spectra have been measured with the help of the same grating spectrometer that was used for the pressure coefficient of the refractive index of this material in ref.7. The grating has a total number of 120000 lines. The monochromator resembles the one described by Barchewitz et al.*); it has a maximum resolution of about 0.2 kayser (= cm-‘), which in the present investigation has been reduced to 1 kayser in order to improve the signal-to-noise ratio. The absolute wave numbers have been calibrated with lines from the neon spectrum and the wave numbers of the absorptivity spectra have been measured relative to these lines with the help of a channelled spectrum of white light through a Fabry-Perot interferometer. This auxiliary spectrum is scanned at the same time as the main spectrum by utilizing a small portion of the entrance and exit slits.
Fig. 1. Optical pressure vessel with specimen exchanger.
238
N. J. TRAPPENIERS
AND R. VETTER
The interferometer has a thickness of nearly 1.5 mm in order to be able to resolve the channels for moderately wide slits. The plates have a reflectance of only 40% and although this results in a poor finesse, this is better matched to the resolving power of the spectrometer than a higher finesse would be and it provides more signal as well. The infrared signal is detected with a PbS photoconductive cell by means of a lock-in amplifier. The monochromator housing is under vacuum so as to reduce complications caused by the refractive index of air in the wave number calibration*) and also to suppress to some degree the atmospheric absorption band that coincides with the germanium absorption edge. The method of double beam in time has been chosen for the absorptivity measurements, necessitating the frequent alternation of two germanium specimens without disturbing the pressure. This was achieved by the pressure vessel shown in fig. 1. It is provided with two quartz windows 9 mm in thickness (C). The vertical steel bar (D) is pressure sealed at either end with a tandem O-ring configuration using rubber O-rings of 90”~shore hardness (E). This piston is raised and lowered with a screw and nut mechanism (F) operated by a crank (G). The specimens (H) are mounted with filler pieces (J) in holes of the specimen exchanger, a third hole allowing the incident flux to be measured for reflection measurements in the region of negligible absorption. The pressure inlet (not shown) is situated at the front wall. Nitrogen gas has been employed as a pressure fluid and the pressure has been measured by a precision Bourdon manometer. Two high purity (35 R cm) single-crystal specimens were used’), 1.4876 cm and 0.5515 cm in thickness respectively and optically polished with diamond abrasive. The ratios @&& of the two transmitted fluxes were read from a recorder chart at regular intervals and reduced to absorptivity, a, by a computer program using eq. (3) in an iterative procedure: di ad = In -L
@2
- In
1 - e2 exp (-atz) 1 - $ exp (-atI)’
(3)
Here d = tz - t, is the thickness difference, and Q is the reflectance, which has been experimentally determined at various pressures. 3. Results. The absorptivity was measured at 1,471, 951 and 952 bar, at 200 points between 5000 and 5250 kayser. The reflectance needed for the correction, was found from the change with pressure of the transmittance r, in absence of absorption : r = (1 - e>m + @I.
(4)
Q decreased from 34% to 29% over 1000 bar. This is consistently lower than the reflectance calculated from the refractive indices and their pressure coefficients
GERMANIUM
ABSORPTION
EDGE UNDER
239
PRESSURE
of germanium and nitrogen by 3%. At the low-energy side of the absorption edge there always remained a small apparent residual absorptivity, which varied little with the series of experimental points. It was ascribed to small changes in the ray geometry with pressure and was corrected for by a small adjustment (~2%) of the transmittance values of each series to those of 1 bar. Fig.2 shows three series out of the four series obtained. For clarity the 951-bar series has been omitted as well as the horizontal parts of the high-pressure series. The structure due to the TO and LA phonon-assisted absorption thresholds (for the l-bar series lying at 5050 and 5120 kayser, respectively) is clearly seen at all pressures and also the dependence of the shift upon the level of the absorptivity. a62
1.6 cd
0.66aV
0.64
0.63
photon energy
-
0.6
photon wavenumber 0
6100
woo
-
6200 kayser
+
Fig. 2. High-resolution
absorptivity
spectra of germanium with pressure.
A computer programme was written so as to obtain the pressure shift of EG from the curves. The experimental points a (0,~) at pressure p and wave number u were shifted back by the amount u(p) along the wave-number scale and muhplied by a factor g-‘(p) until the standard deviation calculated for a whole curve with respect to the broken linear curve through the l-bar series became a minimum. Only the points above a = 0.1 cm-l have been utilized and before subjecting them to the shifting procedure 0.1 cm-l was subtracted and the result multiplied by 45100 kayser. The first correction was applied in accordance with the expectation that the TO component has only the horizontal shift. It was assumed that the remainder, a - 0.1 cm-‘, is proportional to the LA component.
240
N. J. TRAPPENIERS AND R. VETTER
Although it contains contributions of both the TO and LO components, this assumption will be shown to be reasonably fulfilled. The multiplication by u takes care of the energy variable in the denominator of eq. (l), and the division by 5100 kayser, the mean ,wave number of the region, is intended to keep the numerical values to the original order of magnitude. The remaining curves are thus assumed to be of the type ah0 =
constant
fLA[fi~ -
(tiWi- !ioJ)~
-c(P)1 = dP)fLA.
(5)
The minimum of the curve of standard deviation versus pressure shift is taken to be the experimental value of the pressure shift. The scatter below a = 0.1 cm-’ was too large to extract a meaningful result from these parts of the curves. TABLEI Pressure shifts of absorptivity Pressure difference (bar)
Shift (kayser)
spectra of germanium
(1 O-6 eV/bar)
Percentage of decline
m3laP
-a In g/ap (10-6/bar)
20.0 f
1.06
5.28 k 6%
3.8%
81
950
42.04 f
1.06
5.48 + 3%
5.6%
59
951
41.08 + 1.06
5.36 + 3%
5.5%
58
410
Weighted mean
5.39
64
The results of this shifting procedure are displayed in table I, where also a small correction of 0.32 kayser has been included for each 0.1 K by which an experimental pressure series exceeded the temperature of the reference series, in accordance with aE,/aT of ref.2. After dividing the corrected shifts by the pressure differences the values of the pressure coefficient 8&-J+ given in the table are obtained. They have been compounded to a weighted mean value 5.39 x 10m6eV]bar with a standard deviation of 0.1 x 10m6 eV/bar. Column 4 gives the percentages with which the ordinates of the high-pressure series had to be increased in order to coincide with the reference curve. From these numbers the relative decrease per bar of the multiplier g(p) can be calculated. A weighted mean of 64 x 10m6/bar is found for this quantity with an error of the order of 20%. 4. Discussion. Although the shape change of the curves of fig.2 may seem small, the pressure shift depends strongly upon the level of the absorptivity chosen
GERMANIUM
ABSORPTION
EDGE UNDER
241
PRESSURE
to measure it: a variation of 20% occurs even in this small region of low absorptivity. So it is clear that careful analysis of the curves into components is needed. In order to find the contribution of the LA component to the total absorptivity over the whole range, the basic shape of this component at low temperature has been taken from ref. 2. After multiplication by the temperature-dependent phonondistribution factor, application of the temperature broadening and division by 2 (the experimental ratio of the LA creation component to the LA annihilation componenP)) it was fitted to the l-bar curve. This fit agrees with the threshold based on ref.2 within the limit of uncertainty, about 7 kayser.
Ge
294K
1 bar
photon wavenumber
residue
0 5000
5100
Fig. 3. Analysis of germanium absorptivity
5200 kayser
curve into components.
As it is expected that the TO component is allowed it is assumed here to be proportional to the LA component but shifted towards the appropriate threshold of course. A proportionality factor of about 4 seemed to yield the right fit to the low absorptivity part below 5100 kayser. From 5098 kayser onwards, the expected threshold of the LO annihilation componenP), the remainder of the total absorptivity minus the contributions of the LA and TO components, should be ascribed to the LO component (see fig.3). The TO and LO components are expected to shift without shape change and the whole change of slope of the curves under pressure should be due to the LA component only. From fig. 3 it is found that the LA component is proportional to the quantity (a = 0.1 cm-‘) in- the ratio 1: 1.5 within 10%. So this quantity will have a pressure-dependent multiplier proportional to that of the LA component alone, with the same ratio. This justi-
N. J. TRAPPENIERS
242
AND R. VETTER
fies the procedure adopted in the preceding section. The pressure coefficient of the multiplier g(p) of the LA component will therefore be - 1.5 x 64 x 10m6/bar. Indicentally, this yields a value of the pressure coefficient of the direct gap E, (I?,,, + I?,,). From g = constant x (timi - fic~)-~ it follows with fro, = E, and for constant distance from the threshold (#iw = E, - constant) that
ag _ -= gap
2
alapva
iio, -
-
fi4 =
tic0
_2
alapw. -
m
(0.805 - 0.645) eV .
Here E,, = 0.805 eV2) is used and the average value 0.645 eV is taken for tiw. With the above experimental value of a In g/ap and with aE&p = 5.39 x 10m6eV/ bar found in the precedirg sectior, (6) yields aE,,jap = (13.1 &- 2) x 1O-6 eV/ bar. This agrees very well with the only actual measurement at the direct absorption edge: aE,/iTp = (13 f 1) x 10m6 eV/ba?). The main result of the shifting procedure is the experimental value of the shift at a = 0.1 cm-‘. This should be equal to the pressure shift of EG, as it is expected that the TO and the LO components have only a horizontal displacement, and because the LA component is just starting at this level. It has been checked that the corrections due to the pressure dependence of EDh,i and of Eex,* in (2) are negligible, amounting probably to less than 1%. The final value of the pressure coefficient, aE,/ap = 5.39 x 10m6 eV/bar, compares very well with the result 5.42 x 10m6eV/bar of the earlier publicationl). Although the latter was obtained with a prism spectrometer of much lower resolution, the wave-number reproducibility of that spectrometer was found to be very good with respect to the photometric accuracy of these experiments. The optical value of aE,/ap of germanium, 5.4 x 10T6 eV/bar, is thus found to agree with the value obtained from resistivity under pressure. For example, Michels et ~1.~) measured 5.30 x lO-‘j eV/bar, employing pressures up to 3000 bar. Paul and BrookslO) found 5.5 x 10e6 eV/bar (at 350 K) up to 15,0OObar, when pressure-independent mobilities and masses had been assumed, which after correction for electron-mobility variation would be “nearly 10% less”. Nathan, Paul and Brooksll) reported (5.0 f 0.5) x lO-‘j eV/bar, up to 30,000 bar. It is concluded that both electrical and optical determinations refer to the same quantity aE,jap, and that there is no reason to prefer the electrical one as giving the more accurate value. Acknowledgement. This investigation is part of the research programme of the Foundation for Fundamental Research on Matter (F.O.M.), which is supported by the Netherlands Organization for the Advancement of Pure Research (Z.W.O.). The authors are indebted to Mr. M. van der Leij for assistance with the measurements.
GERMANIUM
ABSORPTION
EDGE UNDER
PRESSURE
243
REFERENCES
1) Trappeniers, N. J., Vetter, R. and Andrea, J., Physica 37 (1967) 279. 2) McLean, T. P., Progress in Semiconductors, 3) 4) 5) 6) 7) 8) 9) 10) 11) 12)
vol. 5 (Gibson, Kriiger and Burgess eds.), Heywood (London, 1960), p. 53. Paul, W. and Warschauer, D.M., J. Phys. Chem. Solids 5 (1958) 89. Engeler, W.E., Gariinkel, M. and Tiemann, J. J., Phys. Rev. 155 (1967) 693. Cardona, M. and Paul, W., J. Phys. Chem. Solids 17 (1960) 138. Lax, M. and Hopfield, J. J., Phys. Rev. 124 (1961) 115. Trappeniers, N. J., Vetter, R. and De Bruin, H., Physica 45 (1970) 619. Haeusler, C., Comet, Y. and Barchewitz, P., J. Phys. Radium 21 (1960) 809. Michels, A., Van Eck, J., Machlup, S. and Ten Seldam, C.A., J. Phys. Chem. Solid; 10 (1959) 12. Paul, W. and Brooks, H., Phys. Rev. 94 (1954) 1128. Nathan, M.I., Paul, W. and Brooks, H., Phys. Rev. 124 (1961) 391. Vetter, R., “Effect of Pressure on the Optical Properties of Germanium in the Near Infrared”. Thesis, University of Amsterdam 1970.
0 North-Holland Publishing Co.
HIGH-RESOLUTION
MEASUREMENT
ABSORPTION
EDGE UNDER
N. J. TRAPPENIERS
OF THE GERMANIUM PRESSURE*
and R. VETTER’
Van der Waals-laboratorium, Universiteit van Amsterdam, Nederland (208th publication of the Van der Waals Fund)
Received 15 October 1971
Synopsis The pressure coefficient of the fundamental energy gap of germanium, aE&p, has been obtained with a high-resolution grating spectrometer. The value found, when the different phonon components are taken into account, is (5.39 + 0.1) x 1OWjeV/bar. The analysis also yields a value of the pressure coefficient of the direct gap at k = 0: aE,,/ap = (13.1 f 2) x 1O-6 eV/bar.
1. Introduction. The pressure shift of the absorption edge of germanium near 0.67 eV has been investigated by a number of authors. In a previous publicationl) we have given a survey of the results of these investigations from which it was obvious that there is a considerable variation in the data. It was also shown there that from optical measurements a reliable value of the pressure coefficient of the fundamental energy gap &??d/+ can be obtained. This result is confirmed by the present investigation, in which the prism spectrometer of the former experiment has been replaced by a high-resolution grating spectrometer. It leads to the conclusion that there is no discrepancy between the optical and electrical determinations of &Yo/+. In the present work again most of the attention was paid to the low-absorptivity tail of the edge in order to reduce the errors of extrapolation towards the value a = 0 of the absorptivity. In this region a fine structure is known to exi&) due to phonon-assisted indirect transitions from the valencband top to the minimum of the conduction band. This structure was found’) to persist up to the highest pressures of the experiments (1000 bar), but the details are well resolved only with the high-resolution spectrometer. The shape change under pressure of the absorptivity curves below 1 cm-l, which is due to the pressure dependence of the intermediate states of the indirect transitions?), is clearly detected. An attempt has been made to analyse the curves into the different phonon+ More details on this subject may be found in ref. 12. * Present address: Laboratorium voor Metaalkunde, Technische Hogeschool 235
Delft.
236
N. J. TRAPPENIERS
AND
R. VETTER
assisted components, which is important for a proper interpretation of the pressure shift in terms of aE,/ap. Moreover, from the shape change a value of the pressure coefficient can be found for E,,, the next-higher valency-to conduction-band transition. It can be shown from the theory of indirect transitions in germanium that the absorptivity, a, may be expressed asz)
where w denotes the frequency of the infrared radiative flux and i runs over the 4 phonon components both in annihilation and creation; each Ci is a constant containing density of states effective masses, refractive index, and mass density of germanium and the energies of the respective phonons; iW:b denotes the combined dipole and lattice scattering elements; fioi is the energy difference to the intermediate state;fi is a basic shape function, which is assumed to contain no other pressure-dependent parameters than E,i; Eti is the threshold energy of the respective phonon components, containing the phonon energy Ephef and the exciton binding energy Eex,i in the expression
gi is defined by the last identity of (l), and it is seen to constitute a pressuredependent multiplier. Most of the factors in this multiplier have a relative pressure coefficient of the order of lob6 per bar. The pressure dependence of the matrix elements is unknown. However, in the case of the longitudinal acoustic (LA) phonon, all these pressure coefficients are assumed to be small compared with that of (kiwi - fi~)~. The LA annihilation transition is known to proceed through the rzS conduction-band minimum (see, e.g., ref.4), which lies 0.805 eV above the valency-band maximum at rz5, (ref.2), and has a pressure coefficient aE,/ap of (13 f 1) x 10m6 eV/bar (ref. 5). This will change the multiplier by about 100 x 10S6 per bar. The transverse optical (TO) phonon component proceeds through the LJP intermediate state4). This is separated from the L1 conduction-band minimum by 2.1 eV, which is the major factor in reducing the effect of the multiplier again to 10m6 per bar in the TO case. The longitudinal optical (LO) phonon component is expected to be forbidden”), but from an argument similar to the one given by Lax and Hopfield6) it should become allowed through a A, intermediate state when the transition moves away from the threshold. In that case the LO component also has a nearly pressure-independent multiplier. The absorptivity curves below 1 cm-l consist of the TO, LO, and LA annihilation components, all of which will be submitted to the same shift of the threshold under pressure, as E,,h,i and Eex,* in (2) are found to have negligible pressure coefficients; however, only the first two components will shift purely
GERMANIUM
ABSORPTION
EDGE UNDER
PRESSURE
231
horizontally and the LA annihilation component will have a multiplier that decreases with pressure, thus complicating the measurement of a&/~$. 2. Experimental. The germanium absorptivity spectra have been measured with the help of the same grating spectrometer that was used for the pressure coefficient of the refractive index of this material in ref.7. The grating has a total number of 120000 lines. The monochromator resembles the one described by Barchewitz et al.*); it has a maximum resolution of about 0.2 kayser (= cm-‘), which in the present investigation has been reduced to 1 kayser in order to improve the signal-to-noise ratio. The absolute wave numbers have been calibrated with lines from the neon spectrum and the wave numbers of the absorptivity spectra have been measured relative to these lines with the help of a channelled spectrum of white light through a Fabry-Perot interferometer. This auxiliary spectrum is scanned at the same time as the main spectrum by utilizing a small portion of the entrance and exit slits.
Fig. 1. Optical pressure vessel with specimen exchanger.
238
N. J. TRAPPENIERS
AND R. VETTER
The interferometer has a thickness of nearly 1.5 mm in order to be able to resolve the channels for moderately wide slits. The plates have a reflectance of only 40% and although this results in a poor finesse, this is better matched to the resolving power of the spectrometer than a higher finesse would be and it provides more signal as well. The infrared signal is detected with a PbS photoconductive cell by means of a lock-in amplifier. The monochromator housing is under vacuum so as to reduce complications caused by the refractive index of air in the wave number calibration*) and also to suppress to some degree the atmospheric absorption band that coincides with the germanium absorption edge. The method of double beam in time has been chosen for the absorptivity measurements, necessitating the frequent alternation of two germanium specimens without disturbing the pressure. This was achieved by the pressure vessel shown in fig. 1. It is provided with two quartz windows 9 mm in thickness (C). The vertical steel bar (D) is pressure sealed at either end with a tandem O-ring configuration using rubber O-rings of 90”~shore hardness (E). This piston is raised and lowered with a screw and nut mechanism (F) operated by a crank (G). The specimens (H) are mounted with filler pieces (J) in holes of the specimen exchanger, a third hole allowing the incident flux to be measured for reflection measurements in the region of negligible absorption. The pressure inlet (not shown) is situated at the front wall. Nitrogen gas has been employed as a pressure fluid and the pressure has been measured by a precision Bourdon manometer. Two high purity (35 R cm) single-crystal specimens were used’), 1.4876 cm and 0.5515 cm in thickness respectively and optically polished with diamond abrasive. The ratios @&& of the two transmitted fluxes were read from a recorder chart at regular intervals and reduced to absorptivity, a, by a computer program using eq. (3) in an iterative procedure: di ad = In -L
@2
- In
1 - e2 exp (-atz) 1 - $ exp (-atI)’
(3)
Here d = tz - t, is the thickness difference, and Q is the reflectance, which has been experimentally determined at various pressures. 3. Results. The absorptivity was measured at 1,471, 951 and 952 bar, at 200 points between 5000 and 5250 kayser. The reflectance needed for the correction, was found from the change with pressure of the transmittance r, in absence of absorption : r = (1 - e>m + @I.
(4)
Q decreased from 34% to 29% over 1000 bar. This is consistently lower than the reflectance calculated from the refractive indices and their pressure coefficients
GERMANIUM
ABSORPTION
EDGE UNDER
239
PRESSURE
of germanium and nitrogen by 3%. At the low-energy side of the absorption edge there always remained a small apparent residual absorptivity, which varied little with the series of experimental points. It was ascribed to small changes in the ray geometry with pressure and was corrected for by a small adjustment (~2%) of the transmittance values of each series to those of 1 bar. Fig.2 shows three series out of the four series obtained. For clarity the 951-bar series has been omitted as well as the horizontal parts of the high-pressure series. The structure due to the TO and LA phonon-assisted absorption thresholds (for the l-bar series lying at 5050 and 5120 kayser, respectively) is clearly seen at all pressures and also the dependence of the shift upon the level of the absorptivity. a62
1.6 cd
0.66aV
0.64
0.63
photon energy
-
0.6
photon wavenumber 0
6100
woo
-
6200 kayser
+
Fig. 2. High-resolution
absorptivity
spectra of germanium with pressure.
A computer programme was written so as to obtain the pressure shift of EG from the curves. The experimental points a (0,~) at pressure p and wave number u were shifted back by the amount u(p) along the wave-number scale and muhplied by a factor g-‘(p) until the standard deviation calculated for a whole curve with respect to the broken linear curve through the l-bar series became a minimum. Only the points above a = 0.1 cm-l have been utilized and before subjecting them to the shifting procedure 0.1 cm-l was subtracted and the result multiplied by 45100 kayser. The first correction was applied in accordance with the expectation that the TO component has only the horizontal shift. It was assumed that the remainder, a - 0.1 cm-‘, is proportional to the LA component.
240
N. J. TRAPPENIERS AND R. VETTER
Although it contains contributions of both the TO and LO components, this assumption will be shown to be reasonably fulfilled. The multiplication by u takes care of the energy variable in the denominator of eq. (l), and the division by 5100 kayser, the mean ,wave number of the region, is intended to keep the numerical values to the original order of magnitude. The remaining curves are thus assumed to be of the type ah0 =
constant
fLA[fi~ -
(tiWi- !ioJ)~
-c(P)1 = dP)fLA.
(5)
The minimum of the curve of standard deviation versus pressure shift is taken to be the experimental value of the pressure shift. The scatter below a = 0.1 cm-’ was too large to extract a meaningful result from these parts of the curves. TABLEI Pressure shifts of absorptivity Pressure difference (bar)
Shift (kayser)
spectra of germanium
(1 O-6 eV/bar)
Percentage of decline
m3laP
-a In g/ap (10-6/bar)
20.0 f
1.06
5.28 k 6%
3.8%
81
950
42.04 f
1.06
5.48 + 3%
5.6%
59
951
41.08 + 1.06
5.36 + 3%
5.5%
58
410
Weighted mean
5.39
64
The results of this shifting procedure are displayed in table I, where also a small correction of 0.32 kayser has been included for each 0.1 K by which an experimental pressure series exceeded the temperature of the reference series, in accordance with aE,/aT of ref.2. After dividing the corrected shifts by the pressure differences the values of the pressure coefficient 8&-J+ given in the table are obtained. They have been compounded to a weighted mean value 5.39 x 10m6eV]bar with a standard deviation of 0.1 x 10m6 eV/bar. Column 4 gives the percentages with which the ordinates of the high-pressure series had to be increased in order to coincide with the reference curve. From these numbers the relative decrease per bar of the multiplier g(p) can be calculated. A weighted mean of 64 x 10m6/bar is found for this quantity with an error of the order of 20%. 4. Discussion. Although the shape change of the curves of fig.2 may seem small, the pressure shift depends strongly upon the level of the absorptivity chosen
GERMANIUM
ABSORPTION
EDGE UNDER
241
PRESSURE
to measure it: a variation of 20% occurs even in this small region of low absorptivity. So it is clear that careful analysis of the curves into components is needed. In order to find the contribution of the LA component to the total absorptivity over the whole range, the basic shape of this component at low temperature has been taken from ref. 2. After multiplication by the temperature-dependent phonondistribution factor, application of the temperature broadening and division by 2 (the experimental ratio of the LA creation component to the LA annihilation componenP)) it was fitted to the l-bar curve. This fit agrees with the threshold based on ref.2 within the limit of uncertainty, about 7 kayser.
Ge
294K
1 bar
photon wavenumber
residue
0 5000
5100
Fig. 3. Analysis of germanium absorptivity
5200 kayser
curve into components.
As it is expected that the TO component is allowed it is assumed here to be proportional to the LA component but shifted towards the appropriate threshold of course. A proportionality factor of about 4 seemed to yield the right fit to the low absorptivity part below 5100 kayser. From 5098 kayser onwards, the expected threshold of the LO annihilation componenP), the remainder of the total absorptivity minus the contributions of the LA and TO components, should be ascribed to the LO component (see fig.3). The TO and LO components are expected to shift without shape change and the whole change of slope of the curves under pressure should be due to the LA component only. From fig. 3 it is found that the LA component is proportional to the quantity (a = 0.1 cm-‘) in- the ratio 1: 1.5 within 10%. So this quantity will have a pressure-dependent multiplier proportional to that of the LA component alone, with the same ratio. This justi-
N. J. TRAPPENIERS
242
AND R. VETTER
fies the procedure adopted in the preceding section. The pressure coefficient of the multiplier g(p) of the LA component will therefore be - 1.5 x 64 x 10m6/bar. Indicentally, this yields a value of the pressure coefficient of the direct gap E, (I?,,, + I?,,). From g = constant x (timi - fic~)-~ it follows with fro, = E, and for constant distance from the threshold (#iw = E, - constant) that
ag _ -= gap
2
alapva
iio, -
-
fi4 =
tic0
_2
alapw. -
m
(0.805 - 0.645) eV .
Here E,, = 0.805 eV2) is used and the average value 0.645 eV is taken for tiw. With the above experimental value of a In g/ap and with aE&p = 5.39 x 10m6eV/ bar found in the precedirg sectior, (6) yields aE,,jap = (13.1 &- 2) x 1O-6 eV/ bar. This agrees very well with the only actual measurement at the direct absorption edge: aE,/iTp = (13 f 1) x 10m6 eV/ba?). The main result of the shifting procedure is the experimental value of the shift at a = 0.1 cm-‘. This should be equal to the pressure shift of EG, as it is expected that the TO and the LO components have only a horizontal displacement, and because the LA component is just starting at this level. It has been checked that the corrections due to the pressure dependence of EDh,i and of Eex,* in (2) are negligible, amounting probably to less than 1%. The final value of the pressure coefficient, aE,/ap = 5.39 x 10m6 eV/bar, compares very well with the result 5.42 x 10m6eV/bar of the earlier publicationl). Although the latter was obtained with a prism spectrometer of much lower resolution, the wave-number reproducibility of that spectrometer was found to be very good with respect to the photometric accuracy of these experiments. The optical value of aE,/ap of germanium, 5.4 x 10T6 eV/bar, is thus found to agree with the value obtained from resistivity under pressure. For example, Michels et ~1.~) measured 5.30 x lO-‘j eV/bar, employing pressures up to 3000 bar. Paul and BrookslO) found 5.5 x 10e6 eV/bar (at 350 K) up to 15,0OObar, when pressure-independent mobilities and masses had been assumed, which after correction for electron-mobility variation would be “nearly 10% less”. Nathan, Paul and Brooksll) reported (5.0 f 0.5) x lO-‘j eV/bar, up to 30,000 bar. It is concluded that both electrical and optical determinations refer to the same quantity aE,jap, and that there is no reason to prefer the electrical one as giving the more accurate value. Acknowledgement. This investigation is part of the research programme of the Foundation for Fundamental Research on Matter (F.O.M.), which is supported by the Netherlands Organization for the Advancement of Pure Research (Z.W.O.). The authors are indebted to Mr. M. van der Leij for assistance with the measurements.
GERMANIUM
ABSORPTION
EDGE UNDER
PRESSURE
243
REFERENCES
1) Trappeniers, N. J., Vetter, R. and Andrea, J., Physica 37 (1967) 279. 2) McLean, T. P., Progress in Semiconductors, 3) 4) 5) 6) 7) 8) 9) 10) 11) 12)
vol. 5 (Gibson, Kriiger and Burgess eds.), Heywood (London, 1960), p. 53. Paul, W. and Warschauer, D.M., J. Phys. Chem. Solids 5 (1958) 89. Engeler, W.E., Gariinkel, M. and Tiemann, J. J., Phys. Rev. 155 (1967) 693. Cardona, M. and Paul, W., J. Phys. Chem. Solids 17 (1960) 138. Lax, M. and Hopfield, J. J., Phys. Rev. 124 (1961) 115. Trappeniers, N. J., Vetter, R. and De Bruin, H., Physica 45 (1970) 619. Haeusler, C., Comet, Y. and Barchewitz, P., J. Phys. Radium 21 (1960) 809. Michels, A., Van Eck, J., Machlup, S. and Ten Seldam, C.A., J. Phys. Chem. Solid; 10 (1959) 12. Paul, W. and Brooks, H., Phys. Rev. 94 (1954) 1128. Nathan, M.I., Paul, W. and Brooks, H., Phys. Rev. 124 (1961) 391. Vetter, R., “Effect of Pressure on the Optical Properties of Germanium in the Near Infrared”. Thesis, University of Amsterdam 1970.