Physica B 184 (1993) 216-220 North-Holland
Intersubband scattering in GaAs/A1 x Ga 1- x As heterostructures W. d e L a n g e , F . A . P B l o m , P.J.. v a n H a l l , P.M. K o e n r a a d a n d J . H . W o l t e r Department of Physics, Eindhoven University of Technology, The Netherlands We present measurements of both transport and quantum mobility in a GaAs/A1GaAs heterostructure as a function of the electron concentration, in a range where the second subband becomes occupied. From these measurements we observe a drop in the mobilities at the onset of the second subband. The second subband influences the electrical transport properties by intersubband scattering and screening effects. The individual contribution of these effects determines the height and sign of the step in the mobility and depends on the spatial overlap between the wavefunctions of both subbands, as we show with theoretical model calculations.
In GaAs/A10.33Ga0.67As heterostructures with high electron concentrations, there is often m o r e than one subband populated. These higher subbands substantially influence the electrical transport properties of the two-dimensional electron gas ( 2 D E G ) . The role of these intersubband effects have been examined by a n u m b e r of authors [1-5], but their conclusions about the mobility of the second subband c o m p a r e d to that of the lowest-subband mobility are quite contradictory. Some authors [1-3] found the secondsubband mobility to be lower than the firstsubband mobility. Smith et al.  on the other hand found in one of their samples the secondsubband mobility to be the highest one. Fletcher et al.  calculated an increase of the quantum mobility when the second subband became populated, in contrast to their measurements. Therefore we decided to study the magnetotransport properties in a G a A s / A 1 G a A s heterostructure with various degrees of second-subband occupation and c o m p a r e d the experimental results with theoretical model calculations. We will show f r o m our calculations that the kink in the mobility is very sensitive to the overlap in the spatial extension of the wavefunctions. Correspondence to: W. de Lange, Department of Physics, Eindhoven University of Technology, P.O. Box 513; 5600 MB Eindhoven, The Netherlands.
To describe the electrical behaviour of these systems we have to know the electron concentrations and the mobilities of the individual subbands. One has to distinguish between the transport mobility/.tt, which is determined by the total scattering probability weighted by the transferred m o m e n t u m , and a quantum or single-particle mobility /Xq which is determined by the total unweighed scattering probability [6,7]. Ratios of /xt//xq of the order 10 to 100 are presented in the literature . H a r r a n g et al.  showed that /zt//~ q equals 1 if an isotropic mechanism like e.g. surface roughness scattering is dominant. It is obvious that the ratio of transport and quantum mobilities gives important information about the dominant scattering mechanisms. T h e sample we used is a GaAs/Alo.33Gao.67As heterostructure grown by Molecular B e a m Epitaxy ( M B E ) . On top of the (0 0 1) semi-insulating G a A s substrate a 4 ~m undoped G a A s layer has been grown, followed by a 100/~ undoped A10.33Ga0.67As layer, then a 500/~ n-doped m10.33Ga0.67As layer (3.05 x 1018 c m -3 Si) and a 170/~ u n d o p e d G a A s cap layer. F r o m Hall measurements we found an electron concentration of 6.17 x 1011 cm -2 and a mobility of 96000cm2/ V s in the dark and an electron concentration of 10.8 × 1011 c m -2 and a mobility of 139 600 cm2/ V s after illumination at a t e m p e r a t u r e of 4.2 K. All further experiments have also been carried
0921-4526/93/$06.00 (~) 1993 - Elsevier Science Publishers B.V. All rights reserved
W. de Lange et al. / Intersubband scattering in GaAs/AlxGa I _gAS heterostructures
out at low temperature (4.2 K). The second subband is populated at electron concentrations above 6.2 x 1011 c m -a. T o decrease the electron concentration we mounted the sample in a hydrostatic pressure cell. By applying hydrostatic pressures up to 10 kbar we are able to decrease the electron concentration from 6.17 × 1011 cm -z to 2 × 1011 c m -2 in the dark . In order to get a suitable range of the electron concentration to investigate the intersubband effects we applied a hydrostatic pressure of 1.9kbar. At this fixed pressure we can increase the electron concentration in small steps by flashing a red L E D using the persistent photocondtlctivity effect (PPC). The electron concentrations have been derived from the periodicity of the S h u b n i k o v - D e Haas ( S D H ) oscillations. We determined the average of the transport (or Hall) mobility of both subbands from the zero-field resistivity and the total electron concentration (fig. I(A)). The transport mobility shows a fast increase at the lowest electron concentrations, which is caused by neu15.0 > 13.0
e l e c t r o n c o n c . ( 1 0 " cm "~ ) Fig. 1. Measured average transport mobility (A) and quantum mobility of the lowest subband (B) vs total electron concentration.
tralizing negatively charged acceptors by optically generated e - h pairs. At an electron concentration of about 6.2 x 1011 cm -2 the second subband starts being populated, which is easily observed as a kink in the mobility curve. The second subband contains approximately 0.9 x 10 -11 cm -2 at a total electron concentration of 9 x 10 -11 c m -2. By use of a Fourier filtering technique we separated the S D H oscillations of both subbands. The oscillations of one single subband were used to derive the quantum mobility of this subband from the so-called 'Dingle plots', i.e. ln(Ap~x) vs 1/B. In fig. I(B) we show the quantum mobility of the lowest subband versus the total electron concentration in the range of 4 9 x 10 -11cm -2. The quantum mobility in the lowest subband is approximately a factor 10 smaller than the transport mobility as we expect in heterostructures in which coulomb scattering, caused by ionized donors in the AIGaAs, dominates. The quantum mobility shows a large step in contrast to the transport mobility, which is hardly affected when the second subband becomes populated. We also compared our experimental results with theoretical " model calculations. These are based on ionized donor scattering and interface charge scattering, assuming a very reasonable charge density of 8 x 101° cm -2 at the G a A s / A10.33Ga067As interface. We included the remote impurity scattering using the revised formulation of Van Hall . The intersubband scattering is based on the multi-subband formulation of Siggia and Kwok . Electron-electron screening has been taken into account within a random phase approximation. Variational wavefunctions, chosen to have the same spatial extension ( ( z i ) ) as the wavefunctions from selfconsistent calculations , were used in the mobility calculations. The calculated transport mobility shows a small decrease when the second subband becomes populated, just like the measured transport mobility (fig. 2(A)), but the quantum mobility increases in contrast to the measurements (fig. 2(B)). Fletcher et al.  found the same results in their calculations. The reason of this increase in quantum mobility is the very effective screening of small-angle scattering
W. de Lange et al. / lntersubband scattering in GaAs/AlxGa 1_xAs heterostructures 15.0
B > E o
I'J q , O
1.5 E o
e l e c t r o n c o n e . (1 0 " c m "2)
( 1 0 " cm-" )
Fig. 2. Calculated mobility, based on ionized impurity scattering vs total electron concentration: (A) average transport mobility, (B) quantum mobility.
Fig. 3. Calculated mobility, based on ionized impurity scattering and a second constant-scattering mechanism, vs total electron concentration: (A) transport mobility, (B) quantum mobility of the second subband.
in the lowest subband by the second subband. This screening can be suppressed by introducing a second scattering mechanism acting on the second subband only. This second scattering mechanism, which takes into account the contribution of all scattering mechanisms except ionized donor scattering, will be needed because ionized impurity scattering is not the limiting mechanism, as we conclude from the very high second-subband mobilities. If we do this we find decreasing q u a n t u m and transport mobilities if we populate the second subband (fig. 3). The influence of the second-subband occupation on the lowest-subband mobility is determined by a competition between the effects due to an increase of the density-of-states at the Fermi energy and that of an enhancement of the screening. This is shown in fig. 4, where we demonstrate these effects separately. Curve A shows the mobility in the absence of intersubband scattering, but with the screening effects
of the second subband. Curve B shows the mobility if we only include the intersubband scattering, but not the screening effects. Curve C includes all second subband effects and curve D shows the mobility calculated without any 20.0
A = _=
V 0.0 4
e l e c t r o n c o n e . ( 1011 c m "~ )
Fig. 4. Calculations of the mobility. Curves A-D are explained in the text.
W. de Lange et al. / Intersubband scattering in GaAs/AlxGa I xAs heterostructures
second-subband effects. The relative effectiveness of these second-subband effects, and thus the sign and height of the step in the mobility, will be determined by the overlap between the wavefunctions. A change in the overlap between the wavefunctions of the different subbands can be simulated by changing the acceptor concentration. In fig. 5 we show the quantum and transport mobility calculated with different spatial extensions of the second-subband wavefunction. Figure 5 shows that the step in mobility can be controlled this way. In fig. 6 we show how < Z 1 > depends on the acceptor concentration. Experimentally we can measure the mobility with different (Zl) by illuminating the sample with photons which have an energy below the bandgap energy (infrared), instead of red light. Red light will generate e - h pairs which will
220 A v
200 180 160 le+13
, , , i
. . . . . .
NA¢ ~ (cm 3) Fig. 6. Spatial extension of the second subband, (zl) vs acceptor (background) concentration; (z0) remains constant within 1%.
neutralize the acceptors in the depletion layer . Another possibility to change the overlap in the spatial extension between the wavefunctions by use of a (~ack) gate.
Conclusion 0.0 1.6
The effect of a second populated subband in GaAs/Alo.33Gao.67As heterostructures on the quantum and transport mobility is determined by the screening effects of the second subband and intersubband scattering. The contribution of both effects depends strongly on the spatial overlap between the wavefunctions of the different subbands, which is dependent on the background (acceptor) concentration.
electron conc. (10"cm ~)
Fig. 5. Calculated mobility, based on ionized impurity scattering and a second constant-scattering mechanism, vs total electron concentration: (A) transport mobility, (B) quantum mobility of the lowest subband. The different curves are labeled by the spatial extension of the second subband (( z 1) in ,~) with respect to the wavefunction as obtained from self-consistent calculations.
Acknowledgements We want to thank M.R. Leys, W.C. van de Vleuten and P.A.M. Nouwens for the sample growth and preparation. We are also very grateful to L.M. Weegels for his help with the calculation of the self-consistent wavefunctions.
W. de Lange et al. / lntersubband scattering in GaAs/AlxGa ~ xAS heterostructures
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