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Interband scattering in mobility in GaAs-GaAlAs heterostructures
266
Surface
INTERBAND SCATTERING HETEROSTRUCTURES Nguyen
TOAN
THANG
Science 142 (1984) 266-269 North-Holland, Amsterdam
IN MOBILITY IN GaAs-GaAIAs
* and G. FISHMAN
Groupe de Ph,wique des Soltda de I%NS, Unruersltk Purrs VII, 2 Place Jusswu, F-75251
Pans Cedex OS, Frunce
and B. VINTER Thomson -CSF, Received
Domaine de Corheuille, BP IO, F- 91401 Orsqv, Frunce
8 July 1983; accepted
for publication
5 September
1983
The energy levels and the wave-functions of GaAs-Ga, _,rAl ,As heterostructures are quite different from those of multiple quantum wells: for an Al concentration below about 0.25 and for standard densities, many subbands are below the Fermi level, some of them are mainly in GaAlAs and the others mainly in GaAs. We present realistic calculations of the zero-temperature mobility in this case as a function of donor concentration and buffer width. The interband scattering lowers the mobility by a factor of three when the spacer thickness is zero but only some tens of percent when the spacer thickness is 100 k.
1. Introduction The influence of spacer width on mobility is now well understood in multiple quantum wells [l-3]. In that case the calculation is made easy because the wave functions which correspond to bound energy levels of a quantum well have a simple shape: the electrons are mainly inside the quantum well, and in fact the calculation is very much simplified if the probability of finding an electron outside the well is assumed negligible, which is reasonable in a quantum well. Because of this shape the mobility calculation taking into account the intersubband scattering is tedious but not difficult. It is worth noting that for usual electronic densities at most the two lowest subbands (so-called “0” and “1”) can be populated. The situation can be quite different in a simple heterostructure if the Al concentration x is below about 0.25, so that freeze-out should not be expected in the GaAlAs. Let us recall briefly the geometry of the samples of interest. * On leave of absence
from Center
for Theoretical
Physics,
Nghia Do. Tu Liem, Hanoi.
0039-6028/84/$03.00 6 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
Vietnam.
N. Tom
Thang et al. / Interhand scatrermg in mobility
267
The donors are located between z = 0 and z = d, (we keep the usual notation of quantum wells) in GaAlAs. Between z = d, and z = d, + d, there is the undoped spacer layer (always in GaAlAs). Beyond z = d, + d, the GaAs layer begins up to the total sample thickness d, + d, + d,. Contrary to the quantum well case, the electrons are in the GaAlAs and in a very narrow channel of the GaAs near the interface between GaAs and GaAlAs. We have taken the conduction band discontinuity AE, = 0.22 eV, d, = 600 k and d, = 0, 50 and 100 A, and for the potential we have taken the boundary conditions that V=0.33eVforz=0,andV=Vb=0.2eVatz=d,+d,+2200P\.Asshown in ref. [4], there can be a lot of energy levels below the Fermi level. The aim of this contribution is to show how a mobility calculation can be made in practice in such a case and to point out the main differences from a quantum well. 2. Method and results
In principle we should take into account all the levels. However, on the one hand only two of them correspond to wave functions mainly centered in GaAs and on the other hand the mobility in GaAlAs is known [5] to be weak in comparison with that of GaAs. Then a calculation without any adjustable parameter treating the two kinds of electrons (in GaAs and in GaAlAs) on the same footing would have no meaning. These reasons lead us to take into account only the electrons for which the wave function is mainly in GaAs assuming that the electrons in GaAlAs give a negligible contribution to the mobility. Furthermore we cannot assume that all donors are active Coulomb scatterers, nor that the number of donors is equal to the number of free electrons in GaAs. This is depicted in fig. 1 and indicates one of the main differences from the quantum well case. In order to calculate the number and distribution of active donors in the GaAlAs we use a classical model in which the calculated values of the potential at the surface of GaAlAs and at the GaAlAs-GaAs interface and the calculated number of electrons in the GaAlAs layer are combined with a Schottky approximation for the charge distribution to give the thicknesses of the two depletion layers at the surface and at the doped GaAlAs/undoped spacer layer interface. On the other hand the full quantum mechanical calculation gives the Fermi wave-vector needed to calculate the mobility. With these reasonable approximations the calculation of the zero-temperature mobility proceeds as for quantum wells as described in ref. [3]. However, for quantum wells simple analytic approximations of the subband wave functions can reduce the calculation of the scattering times r0 and r, of subbands 0 and 1 to simple one-dimensional integrals. Unfortunately such simplifications do not seem possible here especially for the first excited subband. In fact, we find that when only one subband is populated, even very simple approximations (such as &functions) give almost correct results, whereas
N. Tom Thung et al. / lnrerbund scatrerrng rn mobrlrg
268
20
d,(i)
t
30
60
90
120 (10"cni")
30
Nd d2
Fig. 1. Density N of the GaAs part of electrons number Nddz of donors in GaAlAs. with undoped A.
60
90
I
120 (10°C~-21
Nd d2
belonging to the channel as a function of total spacer layer thickness d, as parameter. d, = 600
Fig. 2. Calculated mobility of electrons belonging to the channel as a function of total number of donors in GaAIAs. Full curves: intersubband scattering included; dashed curves: mobility of lowest subband only. For d, = 100 A at the highest doping three channel subbands are populated.
when two subbands are occupied, simple approximations lead to erroneous results. Our results are shown in fig. 2 which exhibits the mobility as a function of total donor density (not as a function of electron density) with spacer width d, as a parameter. Comparing with similar results in quantum wells we note that: (i) these results have the same qualitative features, especially the relative jumps of the mobility at the onset of population of the first excited subband are decreasing when the spacer thickness is increasing; (ii) these jumps do not occur for the same value of doping. If we compare our results with the experimental data of ref. [6], we do not find the same electron transfer [4]; this weakens the relevance of a direct comparison between the measured and calculated mobility. In conclusion, we have pointed out the conceptual differences between a mobility calculation in a multiple quantum well system and a simple heterojunction structure of finite size when more than one subband is taken into account. We have shown how reasonable approximations can simplify the problem and have presented results of such a calculation.
References [I] S. Mori and T. Ando, J. Phys. Sot. Japan 48 (1980) 865. [Z] H.L. Stiirmer. A. Pinszuk, A.C. Gossard and W. Wiegmann,
Appl. Phys. Letters 38 (1981) 691.
N. Tom
Thang et al. / Interband scattering in mobility
[3] G. Fishman and D. Calecki, Physica 117/118B (1983) 744; G. Fishman and D. Calecki, to be published. [4] B. Vinter, Surface Sci. 142 (1984) 452. [5] E.g., T. Ishikawa, J. Saito, S. Sasa and S. Hiyamizu, Japan. J. Appl. Phys. 21 (1982) L675 [6] H.L. StGrmer. A.C. Gossard and W. Wiegmann, Solid State Commun. 41 (1982) 707.
269
Surface
INTERBAND SCATTERING HETEROSTRUCTURES Nguyen
TOAN
THANG
Science 142 (1984) 266-269 North-Holland, Amsterdam
IN MOBILITY IN GaAs-GaAIAs
* and G. FISHMAN
Groupe de Ph,wique des Soltda de I%NS, Unruersltk Purrs VII, 2 Place Jusswu, F-75251
Pans Cedex OS, Frunce
and B. VINTER Thomson -CSF, Received
Domaine de Corheuille, BP IO, F- 91401 Orsqv, Frunce
8 July 1983; accepted
for publication
5 September
1983
The energy levels and the wave-functions of GaAs-Ga, _,rAl ,As heterostructures are quite different from those of multiple quantum wells: for an Al concentration below about 0.25 and for standard densities, many subbands are below the Fermi level, some of them are mainly in GaAlAs and the others mainly in GaAs. We present realistic calculations of the zero-temperature mobility in this case as a function of donor concentration and buffer width. The interband scattering lowers the mobility by a factor of three when the spacer thickness is zero but only some tens of percent when the spacer thickness is 100 k.
1. Introduction The influence of spacer width on mobility is now well understood in multiple quantum wells [l-3]. In that case the calculation is made easy because the wave functions which correspond to bound energy levels of a quantum well have a simple shape: the electrons are mainly inside the quantum well, and in fact the calculation is very much simplified if the probability of finding an electron outside the well is assumed negligible, which is reasonable in a quantum well. Because of this shape the mobility calculation taking into account the intersubband scattering is tedious but not difficult. It is worth noting that for usual electronic densities at most the two lowest subbands (so-called “0” and “1”) can be populated. The situation can be quite different in a simple heterostructure if the Al concentration x is below about 0.25, so that freeze-out should not be expected in the GaAlAs. Let us recall briefly the geometry of the samples of interest. * On leave of absence
from Center
for Theoretical
Physics,
Nghia Do. Tu Liem, Hanoi.
0039-6028/84/$03.00 6 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
Vietnam.
N. Tom
Thang et al. / Interhand scatrermg in mobility
267
The donors are located between z = 0 and z = d, (we keep the usual notation of quantum wells) in GaAlAs. Between z = d, and z = d, + d, there is the undoped spacer layer (always in GaAlAs). Beyond z = d, + d, the GaAs layer begins up to the total sample thickness d, + d, + d,. Contrary to the quantum well case, the electrons are in the GaAlAs and in a very narrow channel of the GaAs near the interface between GaAs and GaAlAs. We have taken the conduction band discontinuity AE, = 0.22 eV, d, = 600 k and d, = 0, 50 and 100 A, and for the potential we have taken the boundary conditions that V=0.33eVforz=0,andV=Vb=0.2eVatz=d,+d,+2200P\.Asshown in ref. [4], there can be a lot of energy levels below the Fermi level. The aim of this contribution is to show how a mobility calculation can be made in practice in such a case and to point out the main differences from a quantum well. 2. Method and results
In principle we should take into account all the levels. However, on the one hand only two of them correspond to wave functions mainly centered in GaAs and on the other hand the mobility in GaAlAs is known [5] to be weak in comparison with that of GaAs. Then a calculation without any adjustable parameter treating the two kinds of electrons (in GaAs and in GaAlAs) on the same footing would have no meaning. These reasons lead us to take into account only the electrons for which the wave function is mainly in GaAs assuming that the electrons in GaAlAs give a negligible contribution to the mobility. Furthermore we cannot assume that all donors are active Coulomb scatterers, nor that the number of donors is equal to the number of free electrons in GaAs. This is depicted in fig. 1 and indicates one of the main differences from the quantum well case. In order to calculate the number and distribution of active donors in the GaAlAs we use a classical model in which the calculated values of the potential at the surface of GaAlAs and at the GaAlAs-GaAs interface and the calculated number of electrons in the GaAlAs layer are combined with a Schottky approximation for the charge distribution to give the thicknesses of the two depletion layers at the surface and at the doped GaAlAs/undoped spacer layer interface. On the other hand the full quantum mechanical calculation gives the Fermi wave-vector needed to calculate the mobility. With these reasonable approximations the calculation of the zero-temperature mobility proceeds as for quantum wells as described in ref. [3]. However, for quantum wells simple analytic approximations of the subband wave functions can reduce the calculation of the scattering times r0 and r, of subbands 0 and 1 to simple one-dimensional integrals. Unfortunately such simplifications do not seem possible here especially for the first excited subband. In fact, we find that when only one subband is populated, even very simple approximations (such as &functions) give almost correct results, whereas
N. Tom Thung et al. / lnrerbund scatrerrng rn mobrlrg
268
20
d,(i)
t
30
60
90
120 (10"cni")
30
Nd d2
Fig. 1. Density N of the GaAs part of electrons number Nddz of donors in GaAlAs. with undoped A.
60
90
I
120 (10°C~-21
Nd d2
belonging to the channel as a function of total spacer layer thickness d, as parameter. d, = 600
Fig. 2. Calculated mobility of electrons belonging to the channel as a function of total number of donors in GaAIAs. Full curves: intersubband scattering included; dashed curves: mobility of lowest subband only. For d, = 100 A at the highest doping three channel subbands are populated.
when two subbands are occupied, simple approximations lead to erroneous results. Our results are shown in fig. 2 which exhibits the mobility as a function of total donor density (not as a function of electron density) with spacer width d, as a parameter. Comparing with similar results in quantum wells we note that: (i) these results have the same qualitative features, especially the relative jumps of the mobility at the onset of population of the first excited subband are decreasing when the spacer thickness is increasing; (ii) these jumps do not occur for the same value of doping. If we compare our results with the experimental data of ref. [6], we do not find the same electron transfer [4]; this weakens the relevance of a direct comparison between the measured and calculated mobility. In conclusion, we have pointed out the conceptual differences between a mobility calculation in a multiple quantum well system and a simple heterojunction structure of finite size when more than one subband is taken into account. We have shown how reasonable approximations can simplify the problem and have presented results of such a calculation.
References [I] S. Mori and T. Ando, J. Phys. Sot. Japan 48 (1980) 865. [Z] H.L. Stiirmer. A. Pinszuk, A.C. Gossard and W. Wiegmann,
Appl. Phys. Letters 38 (1981) 691.
N. Tom
Thang et al. / Interband scattering in mobility
[3] G. Fishman and D. Calecki, Physica 117/118B (1983) 744; G. Fishman and D. Calecki, to be published. [4] B. Vinter, Surface Sci. 142 (1984) 452. [5] E.g., T. Ishikawa, J. Saito, S. Sasa and S. Hiyamizu, Japan. J. Appl. Phys. 21 (1982) L675 [6] H.L. StGrmer. A.C. Gossard and W. Wiegmann, Solid State Commun. 41 (1982) 707.
269