Magnetoconductivity probe of inhomogeneity in GaAs-GaAlAs heterojunctions

PflYSICA

Physica B 184 (1993) 206-210 North-Holland

Magnetoconductivity probe of inhomogeneity in GaAs-GaA1As heterojunctions M . L a k n m l ".a,~, A . D . C . Grassle" a,)d K . M . H u t c h i n g s a'b, J.P. Bird" a,,~ j . j . H a r r i s b , f a n d C . T . FOXOI1b'g "Physics Division, University of Sussex, Brighton, UK bphilips Research Laboratories, Cross Oak Lane, Redhill, Surrey, UK CClarendon Laboratory, University of Oxford, UK ~Roedean School, Brighton, UK °Institute of Materials Science, University of Tsukuba, Japan flRC for Semiconductor Materials, Imperial College of Science, London, UK gDepartment of Physics, University of Nottingham, UK We have analysed the variation of the amplitude of the magnetoconductivity peaks of the 2DEG in GaAs/AIGaAs heterojunctions as a function of Landau level index number. We interpret our observations in terms of a scattering change as the magnetic field is varied and the size of the orbit dimensions relative to the scale of long-range potential fluctuations. Furthermore, the experimental values of the conductivity peaks are compared with theoretical predictions. Agreement is possible over a limited range of magnetic field and only when a low areal density of scatterers is used.

1. Introduction

T h e S h u b n i k o v - D e Haas ( S D H ) effect has been extensively used in the study of the twodimensional electron gas ( 2 D E G ) in G a A s / A1GaAs heterostructures. However, there has b e e n little comparison of the precise lineshape and p e a k amplitude of the S D H oscillations with theoretical predictions. It is commonly believed that the screened C o u l o m b potential is short-ranged in SiM O S F E T s because the impurities exist just at the Si/SiO 2 interface and roughness of the interface is the dominant scattering mechanism. The magnetoconductive properties of Si-MOSFETs were understood using the concept of a shortrange potential [1]. Conversely, the potential is long-ranged in G a A s - A 1 G a A s heterojunctions because the parent donors are separated from Correspondence to: M. Lakrimi, Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3 PU, UK.

the electron channel by a spacer layer. In addition, the G a A s / A I G a A s interface is smooth because the two materials are lattice-matched. Analysis of' the envelope of the S D H oscillations yields an estimate of the quantum lifetime ~-q which is related to the phase of the wavefunction and accounts equally for small and large angle scattering. ~-q is the true measure of the broadening since it is even affected by small perturbations whereas the transport lifetime z0, which is related to the zero-field mobility, is only sensitive to large-angle scattering. The nature of the scattering potential affects the ratio %/~-q. In the short-range scattering case, such as the SiM O S F E T , this ratio is of order 1 to 2 while in the case of long-range scatterers, remote ionised impurities and alloy disorder, higher ratios are obtained. In the following, we use very-high-mobility samples to study the dependence of the amplitude of the magnetoconductivity peaks as a function of Landau level index number.

0921-4526/93/$06.00 t~) 1993 - Elsevier Science Publishers B.V. All rights reserved

M. Lakrimi et al. / Inhomogeneity in GaAs-GaAlAs heterojunctions

2. Results and analysis

The GaAs/A1GaAs single heterojunctions were grown by MBE [2], and processed as Hall bars with 22 p~m width and 265 p~m separation between the potential probes. The wafer structure consists of a 4 Ixm layer of GaAs grown on a semi-insulating substrate followed by an undoped spacer layer of A10.33Ga0.67As of 200 thick, then 400 A of doped AI 0 33Gao 67As (N D = 1.3 × 1018 cm -3) and finally a 170/~'capping layer of undoped GaAs. The measurements were taken in magnetic fields up to 6 T and at temperatures between 1.6 and 4.2 K. Some typical transport parameters are given in table 1. The r a t i o "J0/Tq is measured to be between 10 and 60 on these very-high-mobility samples (Ix t> 60 m2/V s) [3] in agreement with ref. [4]. It thus follows that the dominant scattering mechanism is by remote ionised impurities. These values are somewhat higher than those obtained (below 10) on lower-mobility structures (Ix ~< 30 m2/V s) [5]. After measuring the diagonal (off-diagonal) resistivities Pxx (Pxy), the longitudinal conductivity O-xx is given by pxx/(p2, + 0 2 ) . The values of trx,-peaks are initially compared with those given by the model of Ando [1] which assumes shortrange scattering at zero temperature and in the regime ~%rq >> 1, where ~o~ is the cyclotron frequency. Now

207

where N L is the Landau level index number and e the electronic charge. The magnetoconductivity peaks of sample A are shown in fig. 1 as a function of N L. It is clear that the dependence does not obey the above relationship. Indeed, the experimental values are lower by a factor of 26 and furthermore show a nonlinear dependence upon N L. It can be seen that, at high magnetic fields (small NL), o'xxpeaks remain nearly constant, in agreement with the observation of ref. [6], while at low magnetic fields (large NL) o'zz-peaks increase very rapidly with decreasing magnetic field (increasing NL). In addition, the data for sample A is plotted for three different carrier densities in fig. 2. It thus becomes apparent that the crossover between the low- and high-field o'x~-peaks is related to the position of the Fermi energy. Considering the possible degree of inhomogeneity of our samples, we model our electron gas system by a geometrical picture similar to that of Halperin [7] whereby the hills and valleys represent the actual distribution of the longrange potential fluctuations in the sample. At low magnetic field intensity B, the cyclotron orbit diameter is greater than the average range of the potential fluctuations and hence electrons are strongly scattered by the hills, a process which has the nature of short-range scattering giving large angles of scattering. It is

2 p e a k

O'xx

e

~2 ~t

(NL + ½)

(1)

A

B C D E

./

[]

Table 1 Typical transport parameters and variation of the cyclotron diameter at the breakpoint. The data is from five different samples. Sample

7

T [K]

Ns [10 ~5m -2]

ix [m2/V s]

Breakpoint [nm]

4.2 4.2 1.7 4.2 1.7 1.7 4.2 1.7 1.7

3.4 4.8 4.8 5.7 5.7 3.7 5.7 5.7 5.7

87.2 128.7 159.0 94.0 190.0 93.4 117.6 136.0 136.0

210 250 420 310 560 350 230 400 400

c

30

20

0 L)

10

,

0

,

i

L

i

i

i

,

,

i

,

,

I0 20 Landau level number

Fig. 1. Plot of crxx-peak versus N L for sample A at 1.7 K. The arrow indicates ¢D¢'rq = 1 and the lines are to guide the eyes.

M. Lakrimi et al. / lnhomogeneity in G a A s - G a A l A s heterojunctions

208

7

20

The model uses a single-subband approximation at zero temperature where an effective areal density of ionised impurities, n~, located in a plane at some distance z i exerts a coulombic force which is screened by the 2DEG. The model is applicable to both the Si-MOSFET (z~ = 0) and to the GaAs-AlxGal_xAs heterojunction (zi ~ 0). Their model consisted of calculating selfconsistently the width FN of the Landau level giving the following transcendental equation:

[]

% 10 7. • O

0

i

I

J

i

0

i 5

4.8xi0 i

i

~a c m - 2 ,

i

i 10

i

i

i

i

i 15

2 ~ e x p ( - 2 z V £ - x)[L°u(x)] 2 ~/N = 2C, J X,r2 + X / ~ / ( w y s ) e x ~ ( x ) ] Z

Landau level n u m b e r F i g . 2. V a r i a t i o n Landau 4.2K.

of the o'x~-peak (sample

level number

for three

different

A) as a function

of

carrier

at

densities

therefore likely that the trxx-peaks will be determined mainly by this scattering process, hence the strong increase of O'xx with N e (eq. (1)). As the magnetic field is increased, the orbit shrinks and can fit better into the valleys (between two neighbouring hills), hence the carriers can sense the effects of the remote scatters on the far side of the spacer layer, a long-range scattering effect capable of producing the small-angle scattering so important in the quantum oscillatory effects. This yields field-independent peak conductivity [8]. This geometrical argument would explain the data of fig. 1. It is believed that at low carrier densities, when the Fermi level lies in the potential valleys of the samples, the breakpoint of the two lines comes at a low N e (high B), but when the Fermi energy level is raised, i.e. at higher carrier density, the valleys are 'flooded' to a greater depth, allowing larger orbits before scattering from the islands of potentials in the 'flood'; it follows that the breakpoint comes at a larger cyclotron orbit as shown in fig. 2 and given in table 1.

3. Theoretical m o d e l and discussion

In addition the experimental values are compared with the predictions of the self-consistent screening model of Murayama and Ando [8].

dx

(2) where L°(x) is the Laguerre polynomial and z = z i / l 8 . T N = FN/Ec is the reduced width, with " E c = e2 / (eoKlB) being some normalising energy. K is the dielectric constant, e 0 the permittivity constant and c i = 2~rnil ~ represents the number of impurities contained in an area of radius the magnetic length l B = V ~ e B ) . The peak conductivity is then given by peak

o-N

exp(-2zx/-£ - x)[L°(x)] 2 2c~ f 2 + "/N X,~ V~/(,rrYu) exp(_x)[LOu(x)]Z x dx.

(3) We have evaluated eq. (3) using several approximations to represent the finite thickness of the ionised impurities and as a plane of scattering centres. Firstly, we used an areal density equal to that of the electrons and located in a single plane at 200 A from the GaAs/AIGaAs interface. Secondly, we used the actual Si-doping concentration located at 200,~, then at 400/~ and finally uniformly distributed through the doped AIGaAs layer. Neither approximation yielded o-xx-peaks close to those found experimentally. A fit was then calculated using the areal density of the scattering centres and their distance from the 2 D E G as variables. One such best fit for sample D between N L = 7 and 10 is shown in fig. 3. This yields a scattering-centre density of

M. Lakrimi et al. / Inhomogeneity in G a A s - G a A l A s heterojunctions

7

<

17 / / / / ,41 /

v / / / • Q)

/

¢~ 12

/ /m /

r. / / / / 0

~

7

r"

6

t

11 Landau level

i

16 number

Fig. 3. Fit using eq. (3) for sample D. T h e dashed line represents the trxx-peak variation following eq. (1) but scaled by 26 for slope comparison to the data. T h e solid line represents the fit using n~ = 1.2 x 10 x3 m -2 and z~ = 3 2 0 A .

1.2 × 1013 m - 2 located at 320/~ from the 2DEG. It was impossible to obtain a good fit over the whole range of N L because the theoretical prediction shows that O-xxdecreases with N L at high N L. This discrepancy is due to the fact that the number of scattering centres contained within the magnetic length is small and according to this model, the scattering character will be of short range. Since O-xx-peaks are expected to be larger in the short-range scattering regime, we would expect the upward curvature of trxx versus N L as seen in the experimental data of fig. 2. This low areal density of scattering centres is in accordance with several other qualitative models of the electrostatic potential in the GaAsAIGaAs heterojunction system [10]. Localisation effects were discussed in terms of potential fluctuations at the hetero-interface in refs. [9,10] and a model for the quantum Hall effect based on similar inhomogeneities was proposed in ref. [11]. Lassnig [12] suggested that the ionised impurity potentials should not be treated as independent and by taking account of the interference between these potentials the scattering rate should be reduced by orders of magnitude. It was also proposed that scattering from remote ionised impurities is caused by fluctuations from an average distribution [13].

209

We believe that, in the case of fig. 3, the impurities are estimated to be at a greater distance (z i = 320 ~ ) than the spacer layer (200/~) presumably due to migration of the Si-donor atoms inside the AIGaAs-doped layer. Other factors to explain the discrepancy could, for example, be due to the fact that high-mobility samples exhibit a positive background on which the SDH oscillations are superimposed and that is not accounted for by the theory and also not observed in the case of low-mobility samples [3,14]. The Murayama-Ando theory could be improved by including the overlapping between Landau levels. We estimate that the effect of finite temperature is to broaden the effective range of the 'hills' and 'valleys' by only a few nanometers at most. The average 100 nm increase from either side (see table 1) is much greater than expected and cannot be explained as a temperature effect only. We believe that this increasing separation with increasing mobility is a consequence of reduced scattering to acoustic phonons [15]. We did not fully study these dependences since it is very difficult to separate them. Finally, the shape of the sample is known to have an influence on the measured conductivity. Kawaji [16] discussed comparisons between theoretical and experimental O-xx and O-~y. It emerged that O'xx is always smaller in a long Hall bar geometry than in a Corbino disk shape which yields values in very good agreement with the theory. Another complication, in the case of Hall bars, is the distribution of the Hall potential within and in between plateaux [17]. In a plateau region, the Hall potential distribution is well described by edge charge whilst in between plateaux, it is uniformly distributed across the width of the sample with a lesser contribution from edge effects. The tensorial relation would thus not yield the correct conductivity. In view of these facts, a Corbino disk geometry would have been ideal for measuring O-xx directly. In conclusion, we have shown that the Shubnikov-De Haas effect can in fact be interpreted in terms of a scale of microscopic inhomogeneity of the order of several hundred nanometers. We have compared experimentally determined val-

210

M. Lakrimi et al. / Inhomogeneity in G a A s - G a A I A s heterojunctions

ues o f trxx with t h e o r e t i c a l p r e d i c t i o n s . W e h a v e c a l c u l a t e d t h e effective s c a t t e r i n g - c e n t r e d e n s i t y using t h e m o d e l o f M u r a y a m a a n d A n d o a n d f o u n d t h a t this is m u c h l o w e r t h a n t h e d e n s i t y o f silicon d o n o r s i n c o r p o r a t e d in t h e A 1 G a A s l a y e r a n d t h e i r l o c a t i o n r e m a i n s u n r e s o l v e d . W e believe this w o r k p r o v i d e s q u a n t i t a t i v e c o r r o b o r a t i o n o f t h e q u a l i t a t i v e m o d e l s o f p o t e n t i a l fluct u a t i o n s in 2 D s y s t e m s which h a v e b e e n p u b l i s h e d in t h e l i t e r a t u r e . I n a d d i t i o n , w e r e p o r t for t h e first t i m e s h o r t - r a n g e a n d l o n g - r a n g e b e h a v i o u r s b e i n g p r e s e n t in t h e s a m e d a t a . F i n a l l y , o u r results call for m o r e e x p e r i m e n t a l a n d t h e o r e t i c a l l i n e s h a p e analysis.

Acknowledgements This w o r k was s u p p o r t e d b y the Science a n d E n g i n e e r i n g R e s e a r c h C o u n c i l ( U K ) . W e wish to t h a n k Profs. A n d o a n d M u r a y a m a for useful discussions.

References [1] T. Ando, J. Phys. Soc. Jpn. 37 (1974) 1233.

[2] C.T. Foxon and J.J. Harris, Philips J. Res. 41 (1986) 313. [3] M. Lakrimi, PhD thesis, University of Sussex (UK, 1988). [4] P.T. Coleridge, Phys. Rev. B 44 (1991) 3793. [5] R.G. Mani and J.R. Anderson, Phys. Rev. B 37 (1988) 4299. [6] S. Narita, S. Takeyama, W.B. Luo, S. Hiyamizu, K. Nambu and H. Hashimoto, Surf. Sci. 113 (1982) 301. [7] B.I. Halperin, Sci. Am. 254 (4) (1986) 40. [8] Y. Murayama and T. Ando, Phys. Rev. B 35 (1988) 2252. [9] C.T. Foxon, J.J. Harris, R.G. Wheeler and D.E. Lacklison, J. Vac. Sci. Technol. B 4 (1986) 511. [10] V.M. Airaksinen, J.J. Harris, D.E. Lacklison, R.B. Beall, D. Hilton, C.T. Foxon and S.J. Battersby, J. Vac. Sci. Technol. B 6 (1988) 1151. [11] R. Woltjer, R. Eppenga and M.F.H. Schuurmans, Springer Solid State Ser. 71 (1987) 104. [12] R. Lassnig, Solid State Commun. 65 (1988) 765. [13] P.J. van Hall, T. Klaver and J.H. Wolter, Semicond. Sci. Technol. 3 (1988) 120. [14] T. R6tger, G.J.C.L. Bruls, J.C. Maan, P. Wyder, K. Ploog and G. Weimann, Phys. Rev. Lett. 62 (1989) 90. [15] J.J. Harris, C.T. Foxon, D. Hilton, J. Hewett, C. Roberts and S. Auzoux, Surf. Sci. 229 (1990) 113. [16] S. Kawaji, Surf. Sci. 73 (1978) 46. [17] P.F. Fontein, J.A. Kleinen, F.A.P. Blom, J.H. Wolter, L.J. Giling and C.W.J. Beenakker, Surf. Sci. 263 (1992) 91.