UNIVERSAL RELATION BETWEEN BAND RENORMALIZATION AND CARRIER DENSITY IN TWO-DIMENSIONAL ELECTRON-HOLE PLASMAS

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UNIVERSAL RELATION BETWEEN BAND

RENORMALIZATION AND CARRIER DENSITY IN TWO-DIMENSIONAL ELECTRON-HOLE PLASMAS

G. Tränkle, E. Lach, A. Forchel, C. Ell, H. Haug, G. Weimann, G. Griﬀiths, H. Kroemer, S. Subbanna

To cite this version:

G. Tränkle, E. Lach, A. Forchel, C. Ell, H. Haug, et al.. UNIVERSAL RELATION BE- TWEEN BAND RENORMALIZATION AND CARRIER DENSITY IN TWO-DIMENSIONAL ELECTRON-HOLE PLASMAS. Journal de Physique Colloques, 1987, 48 (C5), pp.C5-385-C5-388.

�10.1051/jphyscol:1987582�. �jpa-00226786�

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JOURNAL DE PHYSIQUE

Colloque C5, suppl6ment au noll, Tome 48, novembre 1987

UNIVERSAL RELATION BETWEEN BAND RENORMALIZATION AND CARRIER DENSITY IN TWO-DIMENSIONAL ELECTRON-HOLE PLASMAS

G. TRANKLE, E. LACK, A. FORCHEL, C. ELL*, H. HAUG*, G. WEIMANN* * , G. GRIFFITHS* *

'

⁽^{l )}, ^H.^KROEMER** * and

s .

^SUBBANNA*^{* *}

4. Physikaslisches Institut, Universitst Stuttgart, Pfaffenwaldring 57, 0-7000 Stuttgart-80, F.R.G.

" ~ n s t i t u t fiir Theoretische Physik, Universitzt Frankfurt, Robert-Mayer-Strasse 8 , 0-6000 Frankfurt-am-Main, F.R.G.

* * Forschungsinstitut der Deutschen Bundespost, Am Kavalleriesand 3 , 0-6100 Darmstadt, F.R.G.

* * * Department of Elec. and Comp. Engineering, University of California, Santa Barbara, C A 93106, U.S.A.

Abstract:

We have investigated the band gap renormalization due to many-body effects in electron hole plasmas in quasi-two-dimensional multiple quantum well structures. A comparison of the data obtained in GaAs/GaAlAs-and GaSb/AlSb- structures show that the band gap shift - if scaled in effective Rydberg units

-

is independent of the material system. Theoretical calculations confirm this new universal relation governing the dependence of band renormalization on plasma density in two-dimensional systems.

Many-body effects in high density electron-hole plasmas (EHP) in semiconductors lead to a renormalization of the fundamental band gap [I]. In three-dimensional (3D) structures experiments [2] and many-body theories [3] give a universal law for the dependence of this renormalization on the plasma density independent of material parameters. The comparison of the band renormalization A E in two-dimensional (2D) structures with values from 3D EHP yielded in the case of the

.s

GaAs/GaAlAs-system a surprising difference: For constant interparticle distances the values of the 3D AE in meV are smaller by a factor of about 3 than the 2D values. In Rydberg units, however, the 3D band renormalization is stronger by about g 1 Ry than the 2D renormalization [4]. Our theoretical considerations showed that this can be explained by the dimensionality dependence of screening effects.

To study the effects of material parameters on the band renormalization in 2D systems we performed systematic high excitation photoluminescence measurements at low temperatures. We used GaAs/GaAlAs-MQW-structures with well widths LZ between 2.1 nm and 8.3 nm (Al-content: 43%) and GaSb/AlSb-MQW-samples with LZ between 5.8 nm and 12.1 nm all grown by MBE. In all samples the barrier widths were large enough to avoid coupling between the wells. We excited the EHP using a pulsed dye laser (in the case of GaAs/GaAlAs; r=10 ns) or a frequency-doubled Nd:YAG-laser (in the case of GaSb/AlSb; ~ = 1 0 0 ns) with excitation powers Pex between 100 w/cm2 and I M W / C ~ ' . The pulse widths were large enough to provide quasi-stationary conditions allowing a quasi-equlibrium description.

Backscattering geometry was used to avoid the observation of stimulated emission.

(''NOW at CSIRO. Dxvision of Radio Physxcs, Epping NSW 2121. Australia

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987582

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C5-386 JOURNAL DE PHYSIQUE

GoSb/AISb In all systems we observe a strong broa- dening and red shift (up to 50 meV) of the plasma emission for increasing excitation in-

I I

tensities. The high energy edges of the spectra

flaten as a consequence of the heating of the carrier system due to a nonresonant excitation of the samples and due to reduced carrier coo- ling rates in 2D structures

1.51.

Figure l depicts as a typical example the plasma spectra of a GaSb/AlSb-MQW-sample with LZ = 7.8 nm.

In order to obtain the plasma density, temperature and renormalized band gap we performed a line-shape analysis in a well- established 2D model [6]: Electrons and holes Tp=70K are distributed in parabolic subbands with a step-like 2D density of states. The subband energies are calculated for a rectangular poten- 0.80 0.85 0.90 tial well with finite barriers on the basis of ab-

Energy (eV) sorption or excitation spectroscopy.

&J

Comparison of experimental spectra with

calculated lineshapes We approximated the realistic nonparabolic valence band dispersion by using the bulk effective masses for the heavy and light holes. By comparison with calculated dispersion relations in the case of GaAsIGaAlAs we showed that these bulk values give a good approximation of the exact values if the well widths of the samples is close to the 2D limit. This is a consequence of the large band filling under high excitation conditions 171. In the case of the highly stressed GaSb/AlSb layers (lattice mismatch 0.65%) we took into account the stress effects and calculated the in plane masses assuming the high stress limit [8].

Due to the quasi-equilibrium conditions we assumed common quasi-chemical potentials for the subbands of the electrons and holes,respectively, and a common temperature for all carriers. Furthermore the model included momentum conservation, constant matrix elements for the transitions between different subbands and a collision broadening in the form suggested by Landsberg [9]. This energy broadening has a maximum at the band edge and vanishes at the chemical potential. It does not affect the line widths of the spectra which are a measure of the plasma densities.

The experimental data and the fitted curves agree in all systems very reasonable [4]. The solid lines of figure 1 are typical results of our lineshape analysis in the GaSbIAlSb samples. From the line-shape fits we obtained the plasma density and the renormalized band edges in the high-density plasma. The band gaps for very low densities can be measured in absorption or excitation spectroscopy taking into account the appropriate exciton binding energies. In the case of the of the GaAsIGaAlAs-samples we took binding energies obtained in magneto-optical measurements [lo]. In the case of the GaSblAlSb-system where the exciton binding energies are already expected to be small they were taken to be negligible because the electron-hole interaction is largely screened even at low excitation intensities by a relatively high background carrier concentration (about 10 cm") [I 11.

m

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In figure 2 we show the dependence of the band renomalization on the carrier density for well widths smaller than 12 nm in the two material systems. In both systems the relation between the plasma density and the band ~enormalization is close to AE an*. This agrees with theoretical expectations [12]. The renormalization reaches values between 25 meV and 50 meV for densities between 4 . 1 0 ~ g cm-2 and 2.10'~ cm-*. In any system we observed no systematic dependence of the band renormalization on the well width. This is a clear evidence that the 2D limit is reached in our samples. The absoiute values of the band renormalization in the 2D systems are much larger than those obtained in 3D structures of comparable materials (Figure 2 contains for example the values obtained in 3D GaAsIGaAlAs-mesa structures [2] and

The appropriately scaled experimental results for the 2D systems are shown in figure 3. Consistent with in GaSb-bulk samples [I31 at comparable densities).

the parabolic approximation of the hole subbands used in the line shape analysis, we calculated the Rydberg energies and the effective Bohr radii using the heavy hole masses of the bulk (high stress limit in Band gap renormalization in quasi-two-dimensional systems

0 -'O-T

-

^-20-

3

E

- ^_

w i

-Lm-40-v w

-50- -60

the case of G a s b l a b ) . These masses represent the complicated valence-band structure best for the

10'2 1 0l2

n2D (crnd2) n2D ( ~ m - ~ )

investigated well widths and densities [7]. The experimental results are compared with numerical calculations of the band gap shrinkage AE in 2D systems and experimental [2] as well as theoretical [3]

results which are well established for 3D systems. g

I ' 1 " I

GaAs/GaAIAs

= 2K

* +**

t

301-1

rO-

-

_0-

-

==.'rlr,

2.1 nm

.

4.1 nm

.;-

+

8.3 nrn

*

^3D

1 I t o I I I

Band-gap shift A E scaled by I I I I I

the 2D and 3D Rydberg energies vs g

*

GaSb 7.0nrn the dimensionless interparticle distan-

ce rs for different 2D and 3D struc- tures (Si(4,l) denotes Si under high uniaxial stress parallel to the [110] axis ^o [14]). The numerical results for a 2D

-

plasma are calculated in a dynamical -lm

RPA with a single plasmon pole

*

^3D-GaAs

approximation. The universal 3D law -4

-

^r Ge (4,2)

is taken from Ref. 3. ⁸Si (6.2)

Si (4.1)

0.6 0.8 1.0 1.2 1.4

'-3

I I I I l l

+

5.8nrn

t r 7.0nrn

-

7.8nm

*

¹^0.0nt-n

-

*

3D

+ 0

- - _{- - -} ^-

-0-

-

%-*-

^v ^;-I

GaSb/AISb v

T = 2K

-

t I I I I I

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C5-388 JOURNAL DE PHYSIQUE

The 2D self-energies have been calculated in the dynamical random-phase approximation (RPA) with a single-plasmon pole approximation [12]. We have shown previously that the results are very similar to those obtained from a Hubbard modified dynamical RPA which takes into account the influence of short-range correlations [4].

Figure 3 clearly shows that in the universal units Rydberg energy and Bohr radius in both 2D systems investigated yield a comparable band gap shrinkage close to the theoretical result. Like in 3D systems band gap renormalization in 2D structures is independent of material properties. There is a universal law governing the dependence of the band gap shrinkage on plasma density. Compared to 3D systems the band renormalization is considerably smaller for the same interparticle distances which can be traced to a reduced efficiency of screening in 2D structures [4].

In summary, we have presented measurements and calculations for the density dependence of the band-gap renormalization in 2D EHP in different material systems. In appropriate Rydberg units there is a universal relation governing this dependence independent of material properties. Scaled in the natural units the band gap shift in the 2D EHP is considerably weaker than in the 3D case, as a result of the reduced efficiency of screening in two dimensions. However, the absolute values of the band-gap shifts in 2D are found to be much larger than the corresponding 3D ones. This is due to the increase of the Rydberg energy by a factor of about four going from 3D to 2D.

We thank M. Pilkuhn for stimulating discussions. The work has been supported by the Deutsche Forschungsgemeinhaft partly under Contract No. Pi-71/20 and partly through the Sonderforschungs- bereich Frankfurt-Darmstadt. Parts of this work were carried out in the framework of the European Joint Optical Bistability program, supported by the Commission of the European Communities.

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