Thermalization of hot electrons in quantum wells

Physica 134B (1985) 305-308 North-Holland,Amsterdam

305

THERMALIZATION

OF HOT ELECTRONS

IN QUANTUM

WELLS

C.H. Yang and S.A. Lyon Department of Electrical Engineering, Princeton University, Princeton, NJ 08544

Abstract W e present a new photoluminescence technique for studying hot electron relaxation in quantum wells. T w o lasers are used to measure the density of photoexcited electrons at energies well above the thermalized distribution. The method has been used to determine the relaxation rate for hot electrons in a GaAs/Al1_xGaxAs multi-quantum well structure with an electron density of 2 . 5 × 1 0 H / c m 2. Electrons appear to "bottleneck" in a suhband, thermalizing about I0 times more slowly (0.03 eV/ps) at the bottom of the subbund than higher up.

A number of recent studies of hot electron effects in

rier density allows one to determine the average rate at

modulation doped quantum wells (QW) have shown that

which the carriers lose their excess energy. Concentrating

most of the electron distribution can be well described by a single temperature 1'2. However, the question arises as to

on electrons and assuming conservation of particles we have

the rate at which carriers injected with a large excess R --

energy lose this kinetic energy. At these high energies both electron-electron and electron-phonon scattering may be important.

In

addition

to

the intrinsic scientific

interest in this problem many current and proposed devices (quantum well lasers, planar doped barrier transistors, etc.) involve the injection of carriers with a large excess kinetic energy. Here we describe a new quasi-continuous photoluminescence technique for measuring the average relaxation rate of highly excited electrons in 2-dimensional(2D) systems. It provides much of the same information as pico- and femto-second experiments3,4 but with the advantage that the experiment is much simpler. We have used this method to measure the thermalization rate of electrons near the bottom of the third subband in a modulation doped multi-quantum well sample. We find that the energy loss rate for electrons at the bottom of the subband is about 0.03eV/ps and increases by an order of magnitude higher up in the subband. This is the first experimental indication of an "intersubband bottleneck" for hot electrons in QW. In a photoluminescence experiment carriers are injected by the exciting light at some energy above the band gap of the semiconductor. If only a small percentage of the pairs recombine as they thermalize to the band edge (nearly always the case), then a measurement of the excitation density and the energy dependence of the car-

0378-4363/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

ninj

AF"

(1)

n(E) where n(E) is the density of electrons with energy E (measured relative to the band edge), ninj is the excitation density per unit time, R is the average energy loss rate of the electrons and AE is the energy range over which n(E) is measured. The difficulty is in determining n(E). We have used the hot luminescence, emitted at energies well above that from the thermalized distribution, to determine the electron density as a function of energy in a GaAs QW.

Quantum well structures are particularly

suited for this method since there is no diffusion of carriers into the bulk of the sample.

For simplicity we

assume here only a single conduction and valence band. The intensity of the hot luminescence at energy hv is simply proportional to the electron and hole densities with this energy separation. I(hu) a n(E)p(hv - E s - E)

(2)

The energy E is uniquely determined by hv for the the case of u single band. In order to find n(E) we need to determine the hole density. Here it is fortunate that the hole mass in GaAs is much larger than the electron mass so that holes with the appropriate wavevector have a low energy and are well described by a Maxwell-Boltzmann distribution. Thus we need only to determine the density of holes at the top of the valence band, Po- The hole density at an energy E h (relative to the band edge) is then

CH. Yang and S.A. Lyon / Thermalization of hot electrons

306

given by

the visible laser on.

The contribution from geminate

(3)

recombination will not change when the infrared excita-

The intensity (per unit energy) of the photoluminescenee

tion is turned off because hu is larger than the photon energy of the infrared beam. Taking the difference

at the peak, Ipeak, is proportional to Po and the density of

between Eq. 5 and 6 gives

p(Eh) = po e Eh/kTe

states, go, at the b o t t o m of the lowest electron subband (the sample is modulation-doped and the electrons are degenerate). Using the peak luminescence intensity and Boltzmann factor to find the hole density, we have n(E) = go [ I(hv) ] eEh/kTe ipeak I

Iv+rir - I v a nv(E)Pir

(7)

The geminate contribution cancels out, and thus we measure electrons from the visible laser recombining with holes from the infrared beam.

The difference is obtained by

chopping the infrared excitation and using lockin tech(4)

niques.

Thus we need only to measure the ratio of two lumines-

We have used the method described above to investi-

cence intensities and to calculate the energy (relative to

gate hot electron thermalization in a modulation-doped

the band edge) of the holes involved in the luminescence

GaAs/All_xGg:As (x = .28) multi-quantum well structure

process in order to determine the absolute electron density

with 15 wells of 150,( width and an electron density(both

as a function of energy. An absolute measurement of the luminescence intensity is not necessary.

The infrared excitation used in these experiments was the

The difficulty in implementing the method outlined above for determining n(E) is that not all of the hot luminescence is described by Eq. 2. There is also a contribution from geminate processes in which an electron recombines with the hole it was originally created with. These electrons and holes are not statistically independent and the holes are not in thermal equilibrium with the other carriers in the Q W as assumed in Eq. 3. Thus the However, we have developed a two-laser technique which allows us to avoid the problem with geminate recombination. One high energy (visible) laser injects hot carriers, while a second (infrared) laser injects cold carWe measure the photoluminescence at a photon

energy, hz/, that is between the photon energies of the two lasers. The visible excitation is continuous while the infrared laser is chopped. With the infrared on we have Iv + ir(hv) ~ nv(E)(Pir + Pv) q- Igem

line of a Kr + laser. For most of the experiments

the 5145A

line of an Ar + laser was the visible excitation,

though a dye laser (6728A) was also used.

The sample

was held at 50K in a variable temperature helium dewar. The luminescence spectra were taken with a double spectrometer, GaAs photomultiplier tube, and photon counting electronics.

We found that geminate recombination

dominates the hot luminescence, and thus as much as component. A computer was used as a digital lockin amplifier in order to provide the necessary stability for long integrations. T h e system was tested by moving the chopper out of the infrared beam and observing that it in fact integrated to zero. Stray infrared light passing through the spectrometer appears as a signal.

Thus an

integration was performed for each point with the visible laser off, and this result was subtracted from the measurement with both lasers in order to determine the true sig-

(.5)

where Iv+ir(h~' ) is the luminescence intensity at h~, with both lasers on, nv(E ) is the electron density generated by the visible excitation, Pir(Pv) is the hole density generated by the infrared (visible) excitation, and Igem is the geminate luminescence. T h e infrared photons have a low enough energy that electrons produced by them are not involved in luminescence processes at hi,. When the infrared excitation is off we have I,(hu) a nv(E)p v + I,e m

7525e(

several hours are needed to measure the non-geminate

above analysis breaks down.

riers.

in the dark and under illumination) of 2 . 5 X 1 0 n c m 2.

(6)

where Iv(ha ) is the luminescence intensity at hi, with only

nal. In Fig. 1 we show an excitation spectrum taken on the sample in order to determine the subband structure. This is necessary in order to find the relationship between the photon energy and the electron and hole energies. A tungsten lamp and monoehromutor were used as an excitation source in measuring this spectrum. The subband strueture is clearly observable but we expect to see only steps at the s u b b a n d s energies. T h e origin of the peaks is

CH. Yang and S.A. Lyon / Thermalization of hot electrons

307

not well understood, though they may arise from exciton effects. Similar peaks have been seen by other investigators s. The subband assignments were made by using an effective mass model for the carriers in a QW and treating the band bending with perturbation theory 6. It is not clear why light-hole subbands do not appear.

u~

(We see

%***

only the lowest light-hole subband.)

i

~

F

EXCIIATIDN

. . . .

t

S P E C [HUM

t

I bb

hh~

| Gb

•

i ~/

I btJ I la~ | 70

171

1.72

P L [NEI~GY ( e V )

FIGURE 2

15,5

1.60

I 6.5

Thermalization rate vs. photon energy, hr. The solid circles are data using a visible laser at 5145,~ for excitation. An excitation wavelength of 6728A was used for the solid squares. The data cover a range of electron energies near the bottom of the 3rd subband

I 70

EXCITATION ENERGY{eV)

FIGURE 1 Excitation spectrum of modulation doped multi-quantum well structure. The luminescence was detected at a wavelength of 8110A (1.53 eV). For the transition assignments, hh indicates a process involving a heavy hole. The subscript indicates both the electron and hole subband (0 is the lowest subband).

in these experiments. Another possible error would come from absorption processes which leave a hole in the split-

In Fig. 2 we show the measured thermalization rate

Again, the data taken with 67282[ excitation, which is at

for electrons in the QW. By comparison with the excitation spectrum it can be seen that these data are for elec-

too low an energy to excite electrons from the split-off

off band.

With the lockin technique we would not see

these carriers but that absorption would change the density of hot electrons injected per incident visible photon.

band (or in the AlxGal_xAs ) show that this process does

The solid

not have a large effect. Errors in the temperature or in

circles are data taken using 5145A and the solid squares were obtained with 6728A . It can be seen that the ther-

determining electron and hole energies would change the

trons near the bottom of the third subband.

realization is the slowest near the bottom of the subband and becomes about an order of magnitude faster at higher energies. Thus it appears that there is a bottleneck for electrons at the bottom of the subband.

This result is

consistent with calculations of intrasubband vs. intersubband scattering by LO phonons7,s.

overall scale but would not significantly change the relative rates. In summary we have developed a new method for measuring hot electron relaxation in QW. The technique is based upon using two lasers to measure the hot luminescence without any contribution from geminate recombination. The density of highly excited electrons

estimate that the error in these measurements is about a

and their average thermalization rate can be obtained from this quasi-cw luminescence measurement. This new

factor of 2, mainly from error in determining the carrier

method has the ability to measure average relaxation

generation rate. Other effects might be expected to cause

rates that would otherwise require femto-second laser techniques.

From the scatter in the data at a single energy we

errors. For example, electron-electron scattering will excite electrons from the Fermi sea up into the bands where they are indistinguishable from optically injected electrons. However, the data for 6728A and 5145)~ excitation are essentially the same, but much more of this "splashing" would be expected for the higher energy. Thus we conclude that "splashing" is not very important

We have used the new method to investigate the relaxation of electrons in a GaAs/Ala_xGaxAs QW.

We

find that the thermalization rate for electrons in the third subband varies from ~0.03 eV/ps near the bottom of the subband to ~ 0 . 3 eV/ps farther up in the subband. These results are the first experimental data indicating that a

308

CH. Yang and S.A. Lyon / Thermalization of hot electrons

bottleneck for electrons exists at the bottom of a subband in a QW.

2.

J.M. Carlson, C.H. Yang, S.A. Lyon and J.M. Worlock, to be published.

The authors would like to thank J.M. Worlock for numerous helpful discussions and for the loan of an Ar + laser and the helium dewar, and thank A.C. Gossard and W. Wiegmann for the multi-quantum well sample. This work was supported in part by the Defense Advanced Research Projects Agency through the Office of Naval Research under contract #N00014-83-K-0739, and by the Ford Motor Co. and the National Science Foundation through the Presidential Young Investigator Program.

3.

C.V. Shank, R.L. Fork, R. Yen, J. Shah, B.I. Greene, A.C. Gossard and C. Weisbuch, Solid State Commun. 47, 981 (1983).

4.

D.J. Erskine, A.J. Taylor and C.L. Tang, Appl. Phys. Lett. 45, 54 (1984).

5.

A. Pinezuk, J. Shah, R.C. Miller, A.C. Gossard and W. Wiegmann, Solid State Commun. 50, 735 (1984).

6.

G. Fishman, Phys. Rev. B 27, 7611 (1983).

7.

B.K. Ridley, J. Phys. C 15, 5899 (1982), and F.A. Riddoch and B.K. Ridley, J. Phys. C 16, 6971

References

1.

J. Shah, A. Pinczuk, H.L. StlJrmer, A.C. Gossard and W. Wiegmann, Appl. Phys. Lett. 44, 322 (1984).

(1983). 8.

P.J. Price, Ann. Phys. (NY) 133, 217 (1981).