- Email: [email protected]
Nucleation of electron-hole-drops in Si
Solid State Communications, Vol. 24, pp. 71—74, 1977.
Pergamon Press.
Printed in Great Britain
NUCLEATION OF ELECTRON—HOLE-DROPS IN Si J. Shah, A.H. Dayem and M. Combescot* Bell Telephone Eaboratories, Holmdel, NJ 07733, U.S.A. (Received 23 February 1977 by A. G. Chynoweth) Time resolved luminescence spectroscopy in Si, for pump intensities very close to threshold for electron—hole-drops (EHD) formation, shows that optically excited carriers first form excitons which then begin to form EHD after a delay. The origin of this delay has been determined. The measured time of formation of EHD varies by approximately four orders of magnitude for a 30% change in pump intensity, providing a direct confirmation of the predictions of the nucleation theory. THE THEORY of homogeneous nucleation of electron— hole-drops (EHD) from an ideal gas of excitons has been developed by Silver [1] and Westervelt [2] by including the finite lifetime of the particles in the standard nucleation theory. This theory has been applied by Westervelt eta!. [3] and Etienne eta!. [4] to explain the experimentally observed “up-going” and “downgoing” thresholds (physically related to lifetime) and to determine the electron—hole-drop surface tension for Ge. One major characteristic of the nucleation phenomena is that the time of formation of drops depends strongly on the degree of supersaturation of the gas. We present here a direct confirmation of this fact for EHD by measuring the growth of EHD luminescence in Si at pump powers (F) very close to the threshold pump power (Fth). We find that for a 30% charge in F, the observed growth times vary by approximately a factor of 10,000. By quantitative comparison with theory we show that formation of critical embryos is the limiting factor for the luminescence rise-time for P very close to P~ h~whereas growth of embryos to macroscopic size is the limiting factor at higher P. We also find that the beginning of the formation of EHD is delayed with respect to the beginning of the excitation pulse and that this delay, which can be hundreds of nanoseconds, depends on the pump intensity. Our measurements show directly that the origin of the delay is very simple: the drops do not begin to form until the exciton density reaches a certain critical value. The experimental arrangement was similar to that reported earlier [5], except for a few details. A crystal of 1000 ~2-cmSi was polished in the form of a Weierstrass over-hemisphere of radius 7 mm. The sample was placed inside a Varitemp dewar and its *
temperature was monitored by a Ge diode thermometer In.soldered to it. A 5145 A Argon laser beam was focused to a spot (‘-‘ 150~umdiameter) at the center of the flat face of the crystal. The luminescence was collected from the spherical face, analyzed by a double spectrometer and detected by a sensitive Sl photomultiplier and gated photon counter system with a response time as short as 15 nsec. All the measurements reported here were performed at 10 ±0.05 K. The cavity dumped Argon laser used for these measurements provided flat pulses of variable width and repetition frequency with a risetime 10 nsec. For the first set of measurements we used 4tzsec wide laser pulses with 50 kHz repetition rate. Figure 1 shows two spectra obtained with a 10 nsec gate on photon counter set at two different delays measured from the beginning of the laser pulse. The first spectrum, at 20 nsec, shows only the TO—LO assisted free exciton (FE) peak whereas the second spectrum, at 200 nsec, shows the broad TO—LO assisted electron—hole-drops (EHD) peak in addition to the FE peak [6]. The data clearly show that the optically excited carriers first form free excitons which then nucleate to form EHD [7]. This is a direct demonstration of the fact that under low excitation conditions the EHD nucleate from an exciton gas. Staehli [8] has reported similar results for Ge. In Fig. 2 we show the time evolution of the FE and EHD luminescence for two pump powers close to the c.w. threshold power ~th = 1.25 mW. These results were obtained with a 4jisec laser pulse and a system response time 30 nsec. We can characterize the EHD luminescence by a “delay” TD and a “risetime” rRT. The “delay” is the time difference between the begining of the laser pulse and the time at which the first EHD signal is observed. TD increases from 50 nsec at 5 mW to 100 nsec at 1.9 mW. It is clear from Fig. 2 that EHD begin to form at a time when the exciton -~
Work performed while on leave from the Groupe de Physique des Solides de l’Ecole Normale Superieure, 24 rue thomond, 75231 Paris. 71
72
NUCLEATION OF ELECTRON—HOLE-DROPS IN Si
8
PHOTON ENERGY(hv) 1.080 1.090
1.070
HF-
I ti ti N N N
~2Onsec —200nsec
z
Lu
z
~ z Lu
xl
U (1)
x.~___j
—S.
z Lu I-.
1~z c_~ ~
..,~_,
9.
113
WAVELENGTH(~m) Fig. I. Luminescence spectra of Si at 10K for two different delays from the beginning of a 4~usecwide laser pulse (a) 20 nsec delay: only excitons and (b) 200 nsec delay: excitons and EHD. Incident peak pump power P = 5mW.
,•/
// /
Ui 0
—.
—
N Ii
Ii ii
N
r
ii
Si 1O°K EHL FE ----CALCULATED •
.
_________________
Si10°K
I
N II
S
, 4
Lu Z
I
1.14
1.15
FE+ 100
PiN
12
(1)6 I
1.16
II
15 )—
0
-
N
>-
H~
Lu U Lu U) —
18 1.100
Si:l0°K’
>-
Vol. 24, No. 1
~
~
‘*.
I
0
2
I
I
4 6 TIME (msec
I
8
10
Fig. 3. Time evolution of EHD and FE for long pulses Pl.4mW.
p150 a,
-5
luminescence reaches a certain value. Thus, this delay originates from the simple fact that a certain FE
100 50
>.
~_-....
~120
0
Z
1
2
5
I
10 20 ~ 100 P(mW)
Lu I-.
FE
~ 100
5mW
Lu
z
80
EHL 5mW
U,
Z
~
60 40
FE 1.9mW P I
20
if ‘th
L_ 0 _______
100
DELAY
EHL 1.9mW
(20~.zsec ——
200
300 TIME
400 (nsec)
500
600
Fig. 2. Time evolution of EHD and FE luminescence in Si for two different P. The spectrometer was set at 1 .097 1 .097 eV for FE and 1.080 eV for EHL curves. Spectrometer resolution = 1.1meV. Zero in time corresponds to the beginning of the 4~seclaser pulse. The delay in the formation of EHD for P = 1.9 mW is 100 nsec as mdicated. This delay is plotted vs Pin the insert.
density has to be present before EHD can form. The insert of Fig. 2 shows how this delay varies with the pump power; we note that TD changes rapidly with P near ~th• The “risetime” is the time during which the amount of electron—hole liquid increases, We know that the rate of formation of EHD depends strongly on the degree of supersaturation. We expect TRT to vary rapidly with P near ~th• From Fig. 2 we note that for P = 1 .9mW (within 50% of the c.w. threshold power) the risetime for EHD is quite fast, 150 nsec. Increasing P to 5 mW decreases this risetime, but not substantially. In order to study this risetime closer to threshold we measured the time evolutions of EHD and FE for = 1.4 mW. For this pump power, no EHD luminescence was detected for 4 I.zsec wide pulses between pulses), even without any gate ( the photontocounter. the keeping pulsewidth 40 msec However, and pulsewhen periodwetoincrease 200 msec, the peak and EHD average powers the same as before, we see measureable luminescence, showing that EHD luminescence needs milliseconds instead of microseconds to rise. Signal to noise considerations prevented us from obtaining the time evolution of EHD by scanning a gate across the 40msec pulsewidth; therefore the following technique was used for obtaining this information. Keeping the peak power, the average power and the
Vol. 24, No. 1
NUCLEATION OF ELECTRON—HOLE-DROPS IN Si
duty factor constant, intensity of EHD luminescence was obtained for many different pulsewidths with the gate on the photon counter equal in width to the laser pulsewidth. The measured intensity vs pulsewidth is shown in Fig. 3 for both EHD and FE. Each point represents an integration of 100 sec and special care was taken to avoid slow drifts in the system. The FE intensity is constant for all measured pulsewidths from 4 psec to 40 msec implying that they reach a constant steady state value in less than 4,.zsec. However, the EHD signal shows a very long risetime. If 1(t) is the true evolution of the luminescence signal, our technique of measurement gives for the intensity of the signal at a given pulsewidth, ~,
w
1(W)
=
w J01 1(t) dt.
~-
Assuming for simplicity [9] 1(t) = 1~(l_e_tft), we can calculate 1(W) for a given rise-time r. The calculated 1(W), for ‘r = 0.65 msec is shown in Fig. 3 as the dashed curve. Since TRT 2.2 x r, we conclude from the fit that TRT of EHD luminescence at 1.4 mW is 1.5 msec which is approximately 10,000 times larger than the observed risetime at 1.9mW. Thus, as one expects from theory [1,2] and experimental results [3,8] in Ge, a very dramatic change in the nucleation rate is observed for a very small change in pump intensity. This is the first direct confirmation in Si of the strong dependence of the nucleation rate on supersaturation of the exciton gas predicted by the theory. We now consider how these results can be quantitatively compared to the nucleation theories [1, 2]. Using detailed rate equations developed by Silver [1] and Westervelt [2], one can obtain computer calculations of the time evolutions of the FE and EHD luminescence. Since of wethe arerisetime, interestedhere onlyain the order of magnitude we can use simplifled approach [10]: the time evolution of the number n of e—h in a drop is given by: dn dt
— —
_____
~(R
______
—
exp kTfluI3)
—
73
where nt (flD) = irn~,r53 (rj~)is the unstable (stable) size (corresponding to dn/dt = 0), no is the density of e—h pairs in a drop and a is a constant given by initial conditions. Equation (2) shows that a drop larger than nt reaches its equilibrium size ~D with a characteristic time 3r t cannot reach 0 but an thatembryo a dropgrows smaller ~ Actually tothan n” byn random processes, and the current of critical embryos passing n” and then able to reach ~D can be estimated as J ~ N 5i~: * (= r~/[n~2/3 RJ) is the time for a critical wnere r~ embryo to collect one exciton and Nt the density of such embryos. Assuming statistical equilibrium between the embryos and the exciton gas, Nt = N t is the difference in the pseudo free energy [1,2,where 10] 1 ehltT A between the gas and the condensed phases of nt pairs. Neglecting the lifetime of the embryos [11] At = *2/3 *1/3 sn /2 and n = s/kT in R where the surface tension a = 8 x 103ergcm2 [12].Jis very sensitive toR, because of the exponential term. We have found experimentally that 30% increase in the excitation changes the risetime of the EHD signal by a factor 1 o~Because of the exponential dependence off on At, a similar change in J is obtained when increasingR by 30 percent from R 1 = 12 to R2 = 16 at 10K [13]. Our experimental results provide therefore a direct confirmation of the theoretical prediction that the rate of formation of drops can change extremely rapidly with an increase of the exciton density close to the excitation threshold. Under these conditions the risetime of the EHI) signal (TRT) is controlled by the nucleation time. On the other hand, far above threshold, where the nucleation current is very large, TRT is controlled by the growth of drops to the macroscopic size, and TRT is expected to saturate to ~ 3r0. Experi. mentally, we found a limiting value of TRT 1.5 x lO~ 7sec. sec while r0 results 10 can be understood without considerAll our ing the complications arising from surface excitation and the resultant diffusion of free excitons. The one exception is the observation that the FE luminescence -
~
‘-~
continues to increase even after the beginning of the EHD signal. The nucleation theory predicts a constant or even a decreasing exciton density after the beginning
To
where r 0 is the lifetime, R is the ratio of the exciton density N1 to the thermodynamical density “I?, ~ at 8sec (= [b.~/TN~]’) is thetocollection (-~ lO 10K) and s is related the surfacetime tension a by s = (8ir/3) (3/4irno)213a. This equation can be solved exactly when the exponential is replaced by the first two terms of its power expansion. This approximation conserves contributions coming from the surface terms, One finds: Ir
—
rt
rt/(rt
_rD)Ir
—
rD
rD/(rD-rt)
=
a et/370
of nucleation. However, if the excitons occupy a larger volume longergas times diffusion, density at exciton willbecause imply a of larger numbera constant of excitons which gives rise to a larger luminescence signal. In conclusion, We have presented time-resolved spectra and time-evolution studies of EHD and FE in Si for pump powers very close to threshold. These measurements show unequivocally that photoexcited carriers first form excitons which then nucleate into EHD. Our data have provided an explanation of the observed delay in the beginning of the formation of
74
NUCLEATION OF ELECTRON—HOLE-DROPS IN Si
EHD. Furthermore, the rate of formation of drops varies by approximately a factor of 10,000 for a very small change in pump intensity, providing the first direct conformation in Si of the large changes in nucleation rate predicted by the theory. Comparison with the theory shows that we have observed two different regimes for the formation of the electron—hole-liquid,
Vol. 24, No. I
namely one in which the time to grow to a macroscopic size is the limiting factor and the other in which the number of critical embryos is the limiting factor for the risetime of EHD luminescence. Acknowledgement We thank A.E. DiGiovanni for technical assistance during these measurements. —
REFERENCES 1. 2. 3.
4.
SILVER R.N.,Phys. Rev. Bli, 1569 (l975);Phys. Rev. B12, 5689 (1975). WESTERVELT R.M., Phys. Status Solidi (b) 74, 727 (1976);Phys. Status Solidi (b) 76,31(1976). WESTERVELT R.M., STAEHLI J.L. & HALLER E.E.,Bull. Am. Phys. Soc. 20,471 (1975); Also, WESTERVELT R.M., Physica Status Solidi (b) 76, 31(1976); WESTERVELT R.M., STAEHLI J.L. HALLER E.E. & JEFFRIES C.D., in Lecture Notes in Physics, Vol. 57, p. 270 (Edited by UETA M. & NISHINA Y.). Springer-Verlag, New York (1976). ETIENNE C., BENOIT A LA GUILLANME & VOOS M.,Phys. Rev. B14, 712 (1976).
5.
SHAH J. & DAYEM A.H., Phys. Rev. Lett. 37, 861 (1976).
6.
Luminescence Spectra of EHL and FE in Si have been investigated by HAMMOND R.B., McGILL T.C. & MAYER J.W., Phys. Rev. BiS, 3566 (1976). The data presented in this paper were taken with pump intensities more than four orders of magnitude lower than in reference [5]. STAEHLI J.L., Phys. Status Solidi (b) 75, 451 (1976).
7. 8. 9. 10. 11.
12. 13.
Our data also fit the more complex form predicted (reference [2]) and observed in Ge (reference [3]). COMBESCOT M. & COMBESCOT R.,Phys. Lett. 56A, 228 (1976); more details will be given in SHAH J., DAYEM A.H. & COMBESCOT M. (to be published). We have not taken into account the rotational and translational energy of the embryo as done in the other theories of EHD nucleation. Inclusion of this energy would modify our expression of J by a prefactor which appears to be unimportant. No experimental value of surface energy s is available for Si. The value quoted is an estimate by Westervelt (reference [2]). 3 erg cm2, J(R2)/J(R) 200) It is to interesting to note this change is veryWe sensitive to this a (for a = 7any x 10 and T(for T= 12K,that J(R2)/J(Rl) 200). feel that makes sophisticated determination of prefactors in J irrelevant.
Pergamon Press.
Printed in Great Britain
NUCLEATION OF ELECTRON—HOLE-DROPS IN Si J. Shah, A.H. Dayem and M. Combescot* Bell Telephone Eaboratories, Holmdel, NJ 07733, U.S.A. (Received 23 February 1977 by A. G. Chynoweth) Time resolved luminescence spectroscopy in Si, for pump intensities very close to threshold for electron—hole-drops (EHD) formation, shows that optically excited carriers first form excitons which then begin to form EHD after a delay. The origin of this delay has been determined. The measured time of formation of EHD varies by approximately four orders of magnitude for a 30% change in pump intensity, providing a direct confirmation of the predictions of the nucleation theory. THE THEORY of homogeneous nucleation of electron— hole-drops (EHD) from an ideal gas of excitons has been developed by Silver [1] and Westervelt [2] by including the finite lifetime of the particles in the standard nucleation theory. This theory has been applied by Westervelt eta!. [3] and Etienne eta!. [4] to explain the experimentally observed “up-going” and “downgoing” thresholds (physically related to lifetime) and to determine the electron—hole-drop surface tension for Ge. One major characteristic of the nucleation phenomena is that the time of formation of drops depends strongly on the degree of supersaturation of the gas. We present here a direct confirmation of this fact for EHD by measuring the growth of EHD luminescence in Si at pump powers (F) very close to the threshold pump power (Fth). We find that for a 30% charge in F, the observed growth times vary by approximately a factor of 10,000. By quantitative comparison with theory we show that formation of critical embryos is the limiting factor for the luminescence rise-time for P very close to P~ h~whereas growth of embryos to macroscopic size is the limiting factor at higher P. We also find that the beginning of the formation of EHD is delayed with respect to the beginning of the excitation pulse and that this delay, which can be hundreds of nanoseconds, depends on the pump intensity. Our measurements show directly that the origin of the delay is very simple: the drops do not begin to form until the exciton density reaches a certain critical value. The experimental arrangement was similar to that reported earlier [5], except for a few details. A crystal of 1000 ~2-cmSi was polished in the form of a Weierstrass over-hemisphere of radius 7 mm. The sample was placed inside a Varitemp dewar and its *
temperature was monitored by a Ge diode thermometer In.soldered to it. A 5145 A Argon laser beam was focused to a spot (‘-‘ 150~umdiameter) at the center of the flat face of the crystal. The luminescence was collected from the spherical face, analyzed by a double spectrometer and detected by a sensitive Sl photomultiplier and gated photon counter system with a response time as short as 15 nsec. All the measurements reported here were performed at 10 ±0.05 K. The cavity dumped Argon laser used for these measurements provided flat pulses of variable width and repetition frequency with a risetime 10 nsec. For the first set of measurements we used 4tzsec wide laser pulses with 50 kHz repetition rate. Figure 1 shows two spectra obtained with a 10 nsec gate on photon counter set at two different delays measured from the beginning of the laser pulse. The first spectrum, at 20 nsec, shows only the TO—LO assisted free exciton (FE) peak whereas the second spectrum, at 200 nsec, shows the broad TO—LO assisted electron—hole-drops (EHD) peak in addition to the FE peak [6]. The data clearly show that the optically excited carriers first form free excitons which then nucleate to form EHD [7]. This is a direct demonstration of the fact that under low excitation conditions the EHD nucleate from an exciton gas. Staehli [8] has reported similar results for Ge. In Fig. 2 we show the time evolution of the FE and EHD luminescence for two pump powers close to the c.w. threshold power ~th = 1.25 mW. These results were obtained with a 4jisec laser pulse and a system response time 30 nsec. We can characterize the EHD luminescence by a “delay” TD and a “risetime” rRT. The “delay” is the time difference between the begining of the laser pulse and the time at which the first EHD signal is observed. TD increases from 50 nsec at 5 mW to 100 nsec at 1.9 mW. It is clear from Fig. 2 that EHD begin to form at a time when the exciton -~
Work performed while on leave from the Groupe de Physique des Solides de l’Ecole Normale Superieure, 24 rue thomond, 75231 Paris. 71
72
NUCLEATION OF ELECTRON—HOLE-DROPS IN Si
8
PHOTON ENERGY(hv) 1.080 1.090
1.070
HF-
I ti ti N N N
~2Onsec —200nsec
z
Lu
z
~ z Lu
xl
U (1)
x.~___j
—S.
z Lu I-.
1~z c_~ ~
..,~_,
9.
113
WAVELENGTH(~m) Fig. I. Luminescence spectra of Si at 10K for two different delays from the beginning of a 4~usecwide laser pulse (a) 20 nsec delay: only excitons and (b) 200 nsec delay: excitons and EHD. Incident peak pump power P = 5mW.
,•/
// /
Ui 0
—.
—
N Ii
Ii ii
N
r
ii
Si 1O°K EHL FE ----CALCULATED •
.
_________________
Si10°K
I
N II
S
, 4
Lu Z
I
1.14
1.15
FE+ 100
PiN
12
(1)6 I
1.16
II
15 )—
0
-
N
>-
H~
Lu U Lu U) —
18 1.100
Si:l0°K’
>-
Vol. 24, No. 1
~
~
‘*.
I
0
2
I
I
4 6 TIME (msec
I
8
10
Fig. 3. Time evolution of EHD and FE for long pulses Pl.4mW.
p150 a,
-5
luminescence reaches a certain value. Thus, this delay originates from the simple fact that a certain FE
100 50
>.
~_-....
~120
0
Z
1
2
5
I
10 20 ~ 100 P(mW)
Lu I-.
FE
~ 100
5mW
Lu
z
80
EHL 5mW
U,
Z
~
60 40
FE 1.9mW P I
20
if ‘th
L_ 0 _______
100
DELAY
EHL 1.9mW
(20~.zsec ——
200
300 TIME
400 (nsec)
500
600
Fig. 2. Time evolution of EHD and FE luminescence in Si for two different P. The spectrometer was set at 1 .097 1 .097 eV for FE and 1.080 eV for EHL curves. Spectrometer resolution = 1.1meV. Zero in time corresponds to the beginning of the 4~seclaser pulse. The delay in the formation of EHD for P = 1.9 mW is 100 nsec as mdicated. This delay is plotted vs Pin the insert.
density has to be present before EHD can form. The insert of Fig. 2 shows how this delay varies with the pump power; we note that TD changes rapidly with P near ~th• The “risetime” is the time during which the amount of electron—hole liquid increases, We know that the rate of formation of EHD depends strongly on the degree of supersaturation. We expect TRT to vary rapidly with P near ~th• From Fig. 2 we note that for P = 1 .9mW (within 50% of the c.w. threshold power) the risetime for EHD is quite fast, 150 nsec. Increasing P to 5 mW decreases this risetime, but not substantially. In order to study this risetime closer to threshold we measured the time evolutions of EHD and FE for = 1.4 mW. For this pump power, no EHD luminescence was detected for 4 I.zsec wide pulses between pulses), even without any gate ( the photontocounter. the keeping pulsewidth 40 msec However, and pulsewhen periodwetoincrease 200 msec, the peak and EHD average powers the same as before, we see measureable luminescence, showing that EHD luminescence needs milliseconds instead of microseconds to rise. Signal to noise considerations prevented us from obtaining the time evolution of EHD by scanning a gate across the 40msec pulsewidth; therefore the following technique was used for obtaining this information. Keeping the peak power, the average power and the
Vol. 24, No. 1
NUCLEATION OF ELECTRON—HOLE-DROPS IN Si
duty factor constant, intensity of EHD luminescence was obtained for many different pulsewidths with the gate on the photon counter equal in width to the laser pulsewidth. The measured intensity vs pulsewidth is shown in Fig. 3 for both EHD and FE. Each point represents an integration of 100 sec and special care was taken to avoid slow drifts in the system. The FE intensity is constant for all measured pulsewidths from 4 psec to 40 msec implying that they reach a constant steady state value in less than 4,.zsec. However, the EHD signal shows a very long risetime. If 1(t) is the true evolution of the luminescence signal, our technique of measurement gives for the intensity of the signal at a given pulsewidth, ~,
w
1(W)
=
w J01 1(t) dt.
~-
Assuming for simplicity [9] 1(t) = 1~(l_e_tft), we can calculate 1(W) for a given rise-time r. The calculated 1(W), for ‘r = 0.65 msec is shown in Fig. 3 as the dashed curve. Since TRT 2.2 x r, we conclude from the fit that TRT of EHD luminescence at 1.4 mW is 1.5 msec which is approximately 10,000 times larger than the observed risetime at 1.9mW. Thus, as one expects from theory [1,2] and experimental results [3,8] in Ge, a very dramatic change in the nucleation rate is observed for a very small change in pump intensity. This is the first direct confirmation in Si of the strong dependence of the nucleation rate on supersaturation of the exciton gas predicted by the theory. We now consider how these results can be quantitatively compared to the nucleation theories [1, 2]. Using detailed rate equations developed by Silver [1] and Westervelt [2], one can obtain computer calculations of the time evolutions of the FE and EHD luminescence. Since of wethe arerisetime, interestedhere onlyain the order of magnitude we can use simplifled approach [10]: the time evolution of the number n of e—h in a drop is given by: dn dt
— —
_____
~(R
______
—
exp kTfluI3)
—
73
where nt (flD) = irn~,r53 (rj~)is the unstable (stable) size (corresponding to dn/dt = 0), no is the density of e—h pairs in a drop and a is a constant given by initial conditions. Equation (2) shows that a drop larger than nt reaches its equilibrium size ~D with a characteristic time 3r t cannot reach 0 but an thatembryo a dropgrows smaller ~ Actually tothan n” byn random processes, and the current of critical embryos passing n” and then able to reach ~D can be estimated as J ~ N 5i~: * (= r~/[n~2/3 RJ) is the time for a critical wnere r~ embryo to collect one exciton and Nt the density of such embryos. Assuming statistical equilibrium between the embryos and the exciton gas, Nt = N t is the difference in the pseudo free energy [1,2,where 10] 1 ehltT A between the gas and the condensed phases of nt pairs. Neglecting the lifetime of the embryos [11] At = *2/3 *1/3 sn /2 and n = s/kT in R where the surface tension a = 8 x 103ergcm2 [12].Jis very sensitive toR, because of the exponential term. We have found experimentally that 30% increase in the excitation changes the risetime of the EHD signal by a factor 1 o~Because of the exponential dependence off on At, a similar change in J is obtained when increasingR by 30 percent from R 1 = 12 to R2 = 16 at 10K [13]. Our experimental results provide therefore a direct confirmation of the theoretical prediction that the rate of formation of drops can change extremely rapidly with an increase of the exciton density close to the excitation threshold. Under these conditions the risetime of the EHI) signal (TRT) is controlled by the nucleation time. On the other hand, far above threshold, where the nucleation current is very large, TRT is controlled by the growth of drops to the macroscopic size, and TRT is expected to saturate to ~ 3r0. Experi. mentally, we found a limiting value of TRT 1.5 x lO~ 7sec. sec while r0 results 10 can be understood without considerAll our ing the complications arising from surface excitation and the resultant diffusion of free excitons. The one exception is the observation that the FE luminescence -
~
‘-~
continues to increase even after the beginning of the EHD signal. The nucleation theory predicts a constant or even a decreasing exciton density after the beginning
To
where r 0 is the lifetime, R is the ratio of the exciton density N1 to the thermodynamical density “I?, ~ at 8sec (= [b.~/TN~]’) is thetocollection (-~ lO 10K) and s is related the surfacetime tension a by s = (8ir/3) (3/4irno)213a. This equation can be solved exactly when the exponential is replaced by the first two terms of its power expansion. This approximation conserves contributions coming from the surface terms, One finds: Ir
—
rt
rt/(rt
_rD)Ir
—
rD
rD/(rD-rt)
=
a et/370
of nucleation. However, if the excitons occupy a larger volume longergas times diffusion, density at exciton willbecause imply a of larger numbera constant of excitons which gives rise to a larger luminescence signal. In conclusion, We have presented time-resolved spectra and time-evolution studies of EHD and FE in Si for pump powers very close to threshold. These measurements show unequivocally that photoexcited carriers first form excitons which then nucleate into EHD. Our data have provided an explanation of the observed delay in the beginning of the formation of
74
NUCLEATION OF ELECTRON—HOLE-DROPS IN Si
EHD. Furthermore, the rate of formation of drops varies by approximately a factor of 10,000 for a very small change in pump intensity, providing the first direct conformation in Si of the large changes in nucleation rate predicted by the theory. Comparison with the theory shows that we have observed two different regimes for the formation of the electron—hole-liquid,
Vol. 24, No. I
namely one in which the time to grow to a macroscopic size is the limiting factor and the other in which the number of critical embryos is the limiting factor for the risetime of EHD luminescence. Acknowledgement We thank A.E. DiGiovanni for technical assistance during these measurements. —
REFERENCES 1. 2. 3.
4.
SILVER R.N.,Phys. Rev. Bli, 1569 (l975);Phys. Rev. B12, 5689 (1975). WESTERVELT R.M., Phys. Status Solidi (b) 74, 727 (1976);Phys. Status Solidi (b) 76,31(1976). WESTERVELT R.M., STAEHLI J.L. & HALLER E.E.,Bull. Am. Phys. Soc. 20,471 (1975); Also, WESTERVELT R.M., Physica Status Solidi (b) 76, 31(1976); WESTERVELT R.M., STAEHLI J.L. HALLER E.E. & JEFFRIES C.D., in Lecture Notes in Physics, Vol. 57, p. 270 (Edited by UETA M. & NISHINA Y.). Springer-Verlag, New York (1976). ETIENNE C., BENOIT A LA GUILLANME & VOOS M.,Phys. Rev. B14, 712 (1976).
5.
SHAH J. & DAYEM A.H., Phys. Rev. Lett. 37, 861 (1976).
6.
Luminescence Spectra of EHL and FE in Si have been investigated by HAMMOND R.B., McGILL T.C. & MAYER J.W., Phys. Rev. BiS, 3566 (1976). The data presented in this paper were taken with pump intensities more than four orders of magnitude lower than in reference [5]. STAEHLI J.L., Phys. Status Solidi (b) 75, 451 (1976).
7. 8. 9. 10. 11.
12. 13.
Our data also fit the more complex form predicted (reference [2]) and observed in Ge (reference [3]). COMBESCOT M. & COMBESCOT R.,Phys. Lett. 56A, 228 (1976); more details will be given in SHAH J., DAYEM A.H. & COMBESCOT M. (to be published). We have not taken into account the rotational and translational energy of the embryo as done in the other theories of EHD nucleation. Inclusion of this energy would modify our expression of J by a prefactor which appears to be unimportant. No experimental value of surface energy s is available for Si. The value quoted is an estimate by Westervelt (reference [2]). 3 erg cm2, J(R2)/J(R) 200) It is to interesting to note this change is veryWe sensitive to this a (for a = 7any x 10 and T(for T= 12K,that J(R2)/J(Rl) 200). feel that makes sophisticated determination of prefactors in J irrelevant.