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Time-resolved analysis of infrared plasma absorption due to electron-hole-drop in germanium
Solid State Communications,
Vol. 14, pp. 763-765, 1974.
Pergamon Press.
Printed in Great Britain
TIME-RF~OLVED ANALYSIS OF INFRARED PLASMA ABSORPTION DUE TO ELECTRON-HOLE-DROP IN GERMANIUM K. Fujii and E. Otsuka Department of Physics, College of General Education, Osaka University, Toyonaka, Osaka 560, Japan
(Received 21 December 1973 by Y. Toyozawa)
By way of time-resolved analysis of the infrared plasma absorption, the decay profile of the electron-hole-drop in germanium is examined for the temperature range 1.8-4.2 K with the help of an 1-12O laser. In this temperature range, the time decay in absorption intensity is found exponential as long as 80 ~ e c after photoexcitation. The observed temperature dependence of the decay time can be explained by the evaporation theory. It has been necessary, however, to introduce a temperature dependent term into the work function in the manner like ~ = (14 -+ 1) + (0.5 + 0.2)T2K.
FOR GERMANIUM under high excitation, existence of the so-called electron-hole drops, or the metallic phase condensate of excitons, seems to be of little doubt at low temperatures: When the excitation is stopped, the drops start to decay through internal recombination of electron-hole pairs and evaporation of excitons at their surfaces. By means of the luminescence 2 and cyclotron resonance s experiments, one has been able to make some estimation for the recombination time inside and the work function of evaporation. Meanwhile, a steady-state infrared plasma resonance for the drop has yielded a peak located at "" 8.7 meV with the half-width being > 5 meV. 4 The purpose of this communication is to show a result of time-resolution in plasma absorption, which casts some light on the course of studying the decay process of the drop.
The photo-pulse had a width of ~ 1/asec. The light was led to the sample in the cryostat through the guide of a 3 nun ~ silica rod installed along the axis of the laser beam transmission tube. The pure germanium sample was of a rectangular shape (3.9 X 6.3 X 2.3 rnm 3) and the widest face - (111) - was exposed to the radiations. Temperature was varied between 1.8 and 4.2 K. The triggering time for the laser pulse and the gate position of the boxcar integrator were simultaneously sled with reference to the triggering time for the light flash so that the aperture for signal observation as narrow as 1/asec was always kept at the topflat portion of the laser pulse, s Relative infrared absorption was obtained as a function of the time interval between the excitation light and the boxcar gate position. The time decay in absorption thus found is exponential at all temperatures. It becomes long-lived with decreasing temperature. Since we are flashing a xenon lamp over a relatively large specimen, the average size of the produced drops must be small in comparison with the case of excitation by laser light. So long as the drop radii are smaller than the incident laser wavelength (within the germanium crystal) and the effective skin depth, the absorption due to the spherical drops will approximately by given by s
A low temperature plasma measurement at zero magnetic field was performed with the help of a pulsed H20 laser at the wavelength of 119/am 00.5 meV). The response time of the antimony-doped germanium detector (Nv --NA " 1.3 X l0 s4 cm -s) was made as short as less than 5 psec by careful preparation. Accidental falling of the excitation light on the detector was prevented by the use of black polyethylene sheets. Optical excitation was made with a xenon fiash-lamp at the repetition rate of 20 Hz.
- l ]; 763
(1)
764
TIME-RESOLVED ANALYSIS OF INFRARED PLASMA ABSORPTION
where ~(co) is the dielectric constant within a drop divided by that of the crystal, N the number of drops in unit volume and Va the average volume of a drop. T h e cross section of the Rayleigh scattering by a drop is expected small because o f r e dependence, where r is the radius of the drop. We shall write the total number of pairs within drops in unit volume rt At a fixed temperature, ~(6o) is independent of the drop size on account of the constant carrier density therein. Then the observed absorption should effectively be proportional to n. The rate equation for n consists of internal recombination rate, evaporation of the electron-hole pairs and adsorption of free excitons at the surface of a drop; namely,
Vol. 14, No. 8
T(K)
100
A
5
4
3
2
,
I
,
,
50
¢J In
30
20
h = - n / r o - "/n~T2exp ( - O / k B T ) q- bna/Snex; (2) where "r = 4nA( 3/4nnd) ~ , b = nP®x(3147rnd)2~, nex the density of free excitons, Pex the average thermal velocity of free excitons, na the density of electronhole pairs within a drop, A the Richardson constant, re the recombination time for electron-hole pairs in the drop, and ~ the work function in evaporation. In the case when the third term is negligibly small on account of the relatively small density of free excitons, equation (2) can readily be solved, yielding
n/no = [exp (--t/3ro) + 7to T2no 1/3 exp ( - ~ / k B T ) {exp (--t/3ro) -- 1 }]3;
(3)
where no is the n value at t = 0, or at the moment of photoexcitation. For such a short time interval after photoexcitation as satisfying the condition {1/re + 7T2n~ 1Is exp (-dp/knT)}t/3 < 1, the expression (3) can be approximated by a single exponential form like n/no = exp (--t/r) (4) with l / r "~ 1/re + ~/T2noV3 exp ( - ~ / k n T ) . (5) As a matter of fact, single exponential decay character is confirmed at all temperatures as far as ~ 80 ~asec after photoexcitation and the observed decay time r is given in Fig. 1 as a function of inverse temperature. Data fitting below 3.7 K was first crudely tried with the help of (5) and then refined with (3), tentatively assuming re to be independent of temperature in this temperature interval. 7 One then obtains • = (14 -+ 1)
10
' 2
I
,I
I
I
3
l,
5
6
101
T( K -I )
FIG. 1. Variation of the decay time of the plasma absorption with temperature. The solid line is calculated from equation (3) with (I, = (14 + 0.5T 2) K, r e = 67 lasec and -me 1/3 = 7.5 X l0 s see -1K -2. + (0.5 +-0.2)T2K with Zo = (67 -+ 5) usec and no i/3 = (7.5 -+ 0.5) X l0 s sec -l K -2. One might as well interpret the temperature-dependent part in • to be arising from the behavior of the chemical potential for a condensed Fermi gas; namely,/a = tao -- 5 T 2. An elementary statistical theory predicts
= (1/2)(Tr/3na)2/3(m * + mg)k~/hL
(6)
Putting m* = 0.22mo and mg = 0.35mo in (6), we obtain our temperature dependent part of ~b that n a = 1 . 0 - 3.7 × 1016 cm -3. This is somewhat smaller than the normally whispered value ("- 1017 cm-a). 7 Such an apparent 'inconsistency' may be avoided by slightly modifying our simplifying assumptions made so far. Despite the prematurity as a whole, the general method of attack introduced here seems to be not far away from the right direction. A more extended study will be provided in near future.
Acknowledgements - The authors are so much delighted to thank Dr. T. Ohyama for a pleasant discussion.
Vol. 14, No. 8
TIME-RESOLVED ANALYSIS OF INFRARED PLASMA ABSORPTION
765
REFERENCES 1.
See for summary POKROVSKII Ya., Phys. Status Solidi (a) 11,385 (1972).
2.
POKROVSKII Ya.E. and SVISTUNOVA K.I., Fiz. i Tek. Poluprovodnikov 4, 491 (1970), Soviet Phys. Semiconductors 4,409 (1970).
3.
HENSEL J.C., PHILLIPS T.G. and RICE T.M., Phys. Rev. Lett. 30, 227 (1973).
4.
VAVILOV V£., ZAYATS V.A. and MURZIN V.N., Proc. l Oth Int. Conf. Phys. Semiconductors, Cambridge, Mass., 1970, p. 509.
5.
OTSUKA E., FUJII K. and KOBAYASHI K.L.I., Japan J. Appl. Phys. 12, 1600 (1973).
6.
See for example LANDAU L.D. and LIFSHITZ E.M., Elecrrodynamics of Continuous Media, Pergamon Press 1963, Chap. x.
7.
BENOIT A LA GUILLAUME C., VOOS M. and SALVAN F., Phys. Rev. BS, 3079 (1972).
Durch Zeitaufl6sungsanalyse der Ultrarotplasmaabsorption ist das ZerfaUsprofil des Elektron-Loch-Tr~fchens in Germanium im Temperaturintervall 1.8--4.2 K mit Hilfe des H20 Lasers untersucht worden. In diesem Temperaturintervall wird ein exponentielles zeitliches Zerfall der Absorptionsintensitat bis zu einer Zeitdauer yon 80 ~sec nach Photoanregung gefunden. Die beobachtete Temperaturabhangigkeit der Zerfallszeit wird durch die Verdampfungstheorie erkl~rt. Jedoch ist es notwendig ein temperaturabh~ngiges Glied in die Austrittsarbeit wie ~ = (14 + 1) + (0.5 + 0.2)T 2 K einzufOhren.
Vol. 14, pp. 763-765, 1974.
Pergamon Press.
Printed in Great Britain
TIME-RF~OLVED ANALYSIS OF INFRARED PLASMA ABSORPTION DUE TO ELECTRON-HOLE-DROP IN GERMANIUM K. Fujii and E. Otsuka Department of Physics, College of General Education, Osaka University, Toyonaka, Osaka 560, Japan
(Received 21 December 1973 by Y. Toyozawa)
By way of time-resolved analysis of the infrared plasma absorption, the decay profile of the electron-hole-drop in germanium is examined for the temperature range 1.8-4.2 K with the help of an 1-12O laser. In this temperature range, the time decay in absorption intensity is found exponential as long as 80 ~ e c after photoexcitation. The observed temperature dependence of the decay time can be explained by the evaporation theory. It has been necessary, however, to introduce a temperature dependent term into the work function in the manner like ~ = (14 -+ 1) + (0.5 + 0.2)T2K.
FOR GERMANIUM under high excitation, existence of the so-called electron-hole drops, or the metallic phase condensate of excitons, seems to be of little doubt at low temperatures: When the excitation is stopped, the drops start to decay through internal recombination of electron-hole pairs and evaporation of excitons at their surfaces. By means of the luminescence 2 and cyclotron resonance s experiments, one has been able to make some estimation for the recombination time inside and the work function of evaporation. Meanwhile, a steady-state infrared plasma resonance for the drop has yielded a peak located at "" 8.7 meV with the half-width being > 5 meV. 4 The purpose of this communication is to show a result of time-resolution in plasma absorption, which casts some light on the course of studying the decay process of the drop.
The photo-pulse had a width of ~ 1/asec. The light was led to the sample in the cryostat through the guide of a 3 nun ~ silica rod installed along the axis of the laser beam transmission tube. The pure germanium sample was of a rectangular shape (3.9 X 6.3 X 2.3 rnm 3) and the widest face - (111) - was exposed to the radiations. Temperature was varied between 1.8 and 4.2 K. The triggering time for the laser pulse and the gate position of the boxcar integrator were simultaneously sled with reference to the triggering time for the light flash so that the aperture for signal observation as narrow as 1/asec was always kept at the topflat portion of the laser pulse, s Relative infrared absorption was obtained as a function of the time interval between the excitation light and the boxcar gate position. The time decay in absorption thus found is exponential at all temperatures. It becomes long-lived with decreasing temperature. Since we are flashing a xenon lamp over a relatively large specimen, the average size of the produced drops must be small in comparison with the case of excitation by laser light. So long as the drop radii are smaller than the incident laser wavelength (within the germanium crystal) and the effective skin depth, the absorption due to the spherical drops will approximately by given by s
A low temperature plasma measurement at zero magnetic field was performed with the help of a pulsed H20 laser at the wavelength of 119/am 00.5 meV). The response time of the antimony-doped germanium detector (Nv --NA " 1.3 X l0 s4 cm -s) was made as short as less than 5 psec by careful preparation. Accidental falling of the excitation light on the detector was prevented by the use of black polyethylene sheets. Optical excitation was made with a xenon fiash-lamp at the repetition rate of 20 Hz.
- l ]; 763
(1)
764
TIME-RESOLVED ANALYSIS OF INFRARED PLASMA ABSORPTION
where ~(co) is the dielectric constant within a drop divided by that of the crystal, N the number of drops in unit volume and Va the average volume of a drop. T h e cross section of the Rayleigh scattering by a drop is expected small because o f r e dependence, where r is the radius of the drop. We shall write the total number of pairs within drops in unit volume rt At a fixed temperature, ~(6o) is independent of the drop size on account of the constant carrier density therein. Then the observed absorption should effectively be proportional to n. The rate equation for n consists of internal recombination rate, evaporation of the electron-hole pairs and adsorption of free excitons at the surface of a drop; namely,
Vol. 14, No. 8
T(K)
100
A
5
4
3
2
,
I
,
,
50
¢J In
30
20
h = - n / r o - "/n~T2exp ( - O / k B T ) q- bna/Snex; (2) where "r = 4nA( 3/4nnd) ~ , b = nP®x(3147rnd)2~, nex the density of free excitons, Pex the average thermal velocity of free excitons, na the density of electronhole pairs within a drop, A the Richardson constant, re the recombination time for electron-hole pairs in the drop, and ~ the work function in evaporation. In the case when the third term is negligibly small on account of the relatively small density of free excitons, equation (2) can readily be solved, yielding
n/no = [exp (--t/3ro) + 7to T2no 1/3 exp ( - ~ / k B T ) {exp (--t/3ro) -- 1 }]3;
(3)
where no is the n value at t = 0, or at the moment of photoexcitation. For such a short time interval after photoexcitation as satisfying the condition {1/re + 7T2n~ 1Is exp (-dp/knT)}t/3 < 1, the expression (3) can be approximated by a single exponential form like n/no = exp (--t/r) (4) with l / r "~ 1/re + ~/T2noV3 exp ( - ~ / k n T ) . (5) As a matter of fact, single exponential decay character is confirmed at all temperatures as far as ~ 80 ~asec after photoexcitation and the observed decay time r is given in Fig. 1 as a function of inverse temperature. Data fitting below 3.7 K was first crudely tried with the help of (5) and then refined with (3), tentatively assuming re to be independent of temperature in this temperature interval. 7 One then obtains • = (14 -+ 1)
10
' 2
I
,I
I
I
3
l,
5
6
101
T( K -I )
FIG. 1. Variation of the decay time of the plasma absorption with temperature. The solid line is calculated from equation (3) with (I, = (14 + 0.5T 2) K, r e = 67 lasec and -me 1/3 = 7.5 X l0 s see -1K -2. + (0.5 +-0.2)T2K with Zo = (67 -+ 5) usec and no i/3 = (7.5 -+ 0.5) X l0 s sec -l K -2. One might as well interpret the temperature-dependent part in • to be arising from the behavior of the chemical potential for a condensed Fermi gas; namely,/a = tao -- 5 T 2. An elementary statistical theory predicts
= (1/2)(Tr/3na)2/3(m * + mg)k~/hL
(6)
Putting m* = 0.22mo and mg = 0.35mo in (6), we obtain our temperature dependent part of ~b that n a = 1 . 0 - 3.7 × 1016 cm -3. This is somewhat smaller than the normally whispered value ("- 1017 cm-a). 7 Such an apparent 'inconsistency' may be avoided by slightly modifying our simplifying assumptions made so far. Despite the prematurity as a whole, the general method of attack introduced here seems to be not far away from the right direction. A more extended study will be provided in near future.
Acknowledgements - The authors are so much delighted to thank Dr. T. Ohyama for a pleasant discussion.
Vol. 14, No. 8
TIME-RESOLVED ANALYSIS OF INFRARED PLASMA ABSORPTION
765
REFERENCES 1.
See for summary POKROVSKII Ya., Phys. Status Solidi (a) 11,385 (1972).
2.
POKROVSKII Ya.E. and SVISTUNOVA K.I., Fiz. i Tek. Poluprovodnikov 4, 491 (1970), Soviet Phys. Semiconductors 4,409 (1970).
3.
HENSEL J.C., PHILLIPS T.G. and RICE T.M., Phys. Rev. Lett. 30, 227 (1973).
4.
VAVILOV V£., ZAYATS V.A. and MURZIN V.N., Proc. l Oth Int. Conf. Phys. Semiconductors, Cambridge, Mass., 1970, p. 509.
5.
OTSUKA E., FUJII K. and KOBAYASHI K.L.I., Japan J. Appl. Phys. 12, 1600 (1973).
6.
See for example LANDAU L.D. and LIFSHITZ E.M., Elecrrodynamics of Continuous Media, Pergamon Press 1963, Chap. x.
7.
BENOIT A LA GUILLAUME C., VOOS M. and SALVAN F., Phys. Rev. BS, 3079 (1972).
Durch Zeitaufl6sungsanalyse der Ultrarotplasmaabsorption ist das ZerfaUsprofil des Elektron-Loch-Tr~fchens in Germanium im Temperaturintervall 1.8--4.2 K mit Hilfe des H20 Lasers untersucht worden. In diesem Temperaturintervall wird ein exponentielles zeitliches Zerfall der Absorptionsintensitat bis zu einer Zeitdauer yon 80 ~sec nach Photoanregung gefunden. Die beobachtete Temperaturabhangigkeit der Zerfallszeit wird durch die Verdampfungstheorie erkl~rt. Jedoch ist es notwendig ein temperaturabh~ngiges Glied in die Austrittsarbeit wie ~ = (14 + 1) + (0.5 + 0.2)T 2 K einzufOhren.