Scattering potential for interface roughness scattering

Applied Surface Science 182 (2001) 357±360

Scattering potential for interface roughness scattering B.R. Nag*, Madhumita Das Institute of Radio Physics and Electronics, 92, Acharya Prafulla Chandra Road, Calcutta 700009, India

Abstract Interface roughness scattering potential is computed for the AlAs/GaAs and Ga0.5In0.5P/GaAs systems by using the two models proposed in the literature: the so-called eigenvalue model and the wave function model. Potentials are found to differ by large factors for the AlAs/GaAs system, while for the Ga0.5In0.5P/GaAs system the difference is not very signi®cant. Values of asperity height, D, and correlation length, L, of the roughness are also calculated for the two systems, which give agreement between theory and experiment for the wave function model. It is found that the results for the AlAs/GaAs system cannot be explained by the wave function model if the roughness protrudes into the barrier; the required values of D and L for protrusion into the well also differ from those for the eigenvalue model by large factors. For the Ga0.5In0.5P/GaAs wells, on the other hand, the difference in the values lies between 10 and 20%. # 2001 Elsevier Science B.V. All rights reserved. PACS: 72.10 -d; 72.10.Fk; 72.15 Lh; 72.20 Fr Keywords: Interface roughness; Scattering potential; Electron mobility; AlAs/GaAs; Ga0.5In0.5P/GaAs

Very low-temperature electron mobility in heterostructures of different compositions have been explained to be determined by the interface roughness scattering (IFRS) [1±11]. The IFRS potential has, however, been modelled in two ways. In one model, hereinafter referred as eigenvalue model, the potential is taken to be the same as the change in the energy eigenvalue due to the roughness. In the second model [12], the potential is taken to be equal to the barrier potential or the well potential according to the extent of the roughness, respectively, into the well layer or the barrier layer. The two models give identical results [13] if the barrier potential is very large, the effective masses are the same in the two layers and the asperity height of roughness is small. Most of the theoretical analyses of IFRS-limited mobility has, however, been

done by using the eigenvalue model. Recently, Nag et al. [14] also explained the experimental results for AlAs/GaAs and Ga0.5In0.5P/GaAs systems by using the eigenvalue model. It is of interest to examine whether the results are altered if the wave function model is used, since the condition required for the equivalence of the two models do not apply to these systems. A quantum well is considered to have a thickness L and barrier height V0 and grown along the z-direction. The eigenvalues of energy are obtained by solving the following equation [15]:   kw L tan ˆ r; 2 r ˆ ‰kb mw …1 ‡ 2aw En †Šfkw mb ‰1

En †Šg 1 ; (1)

*

Corresponding author. Tel.: ‡91-33-337-2696; fax: ‡91-33-351-5828. E-mail address: [email protected] (B.R. Nag).

2ab …V0

where mw (mb) are the band-edge effective masses and aw (ab) are the non-parabolicity in the Kane relation

0169-4332/01/$ ± see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 9 - 4 3 3 2 ( 0 1 ) 0 0 4 4 8 - 2

358

B.R. Nag, M. Das / Applied Surface Science 182 (2001) 357±360

for the electrons in the well (barrier) layer. The wave vectors kw and kb are given by 

1=2

2mw En …1 ‡ aw En † ; h2   2mb …V0 En †‰1 ab …V0 kb ˆ h2 

kw ˆ

(2)  En †Š 1=2

;

(3)

h being (1/2p) times Planck's constant. The envelope function is given by c…r; z† ˆ F…z† exp…ik
r†;

(4)

where k is the in-plane component of the wave vector and r the in-plane position coordinate. The z-dependent part of the envelope function, F(z), is given with z ˆ 0 at the midpoint of the well by 8 L > > A cos kw z for jzj  ; > > 2 > >      > < kw L L L for z  ; F…z† ˆ A cos 2 exp kb z 2 2 > >       > > > k L L L w > > ; exp kb z ‡ for z  : A cos 2 2 2 where A2 ˆ



    1 L kw L=2 kw L=2† ‡ sin : ‡ cos2 2 2kw kb

(6)

The scattering potential in the two models are given for the asperity height D(r) by the following expressions: VE ˆ En …L  D†

En

for the eigenvalue model [14] and Z L=2 VW ˆ DEc ‰F…z†Š2 dz L=2D

(7)

VW mb 1 2ab …V0 En † 1 ‡ a w En ˆ

4 ‡ 2aw En VE mw 1 ab …V0 En † (10) for non-parabolic bands. The scattering potential is larger and the mobility is smaller for the wave function model if non-parabolicity is neglected, since mb > mw . On the other hand, when non-parabolicity is included, VW/VE will vary with the well width; the exact nature of variation will depend on the barrier height and the non-parabolicity parameters. Computations have been done for the two potentials and their ratio is plotted in Fig. 1 for the AlAs/GaAs  and Ga0.5In0.5P/GaAs systems for D ˆ 5:67 A. The physical constants are the same as used in Ref. [14]. These are repeated in Table 1 for completeness. We ®nd that the ratio of the two scattering potentials, VW‡/VE‡ and VW /VE have different values. In the Ga0.5In0.5P/GaAs system, VW‡/VE‡ ranges between 0.92 and 0.87 and VW /VE between 1.45 Ê. and 1.21, as the well width changes from 10 to 100 A On the other hand, in the AlAs/GaAs system, VW‡/ VE‡ decreases from 0.77 to 0.08 and VW /VE from 1.7 to 0.67, for the same changes in the well width. We may, hence, conclude that the results for the Ga0.5In0.5P/GaAs system will not be much different for the two models of scattering potential. The difference would be much larger for the AlAs/GaAs system. We have calculated the electron mobility by using the values of VW and VE in the expression given below [14]: mˆ

(8)

for the wave function model [12], where ‡ and subscript correspond to the protrusion of the roughness, respectively, into the barrier layer and the well layer, and DEc is the conduction-band offset. It may be easily veri®ed that for small values of D(r) and large values of V0, VW mb ˆ VE mw

for parabolic bands, and

(9)

jejt…EF † ; m …E0 †

(11)

 2 1 2 pL ˆ …VW † (12) m …E0 †G…L; kF †; t…EF † h3  Z 2p  2 2  1 L kF …1 cos y† 2 S exp G…L; kF † ˆ 2p 0 c 2  …1

cos y† dy;

(13) 

where Sc is the screening factor, m (E0) the in-plane velocity effective mass including the effects of band non-parabolicity and the wave function penetration into the barrier and kF the Fermi wave vector. The

B.R. Nag, M. Das / Applied Surface Science 182 (2001) 357±360

359



Fig. 1. Ratio of scattering potentials VW and VE for different well widths for asperity height D ˆ 5:67 A. Solid line Ð VW‡/VE‡ for AlAs/ GaAs. Dashed line Ð VW /VE for AlAs/GaAs. Dot dashed line Ð VW‡/VE‡ for Ga0.5In0.5P/GaAs. Dotted line Ð VW /VE for Ga0.5In0.5P/ GaAs. Table 1 Physical constants of the materials of the well Material

Effective mass ratio

Non-parabolicity parameter (eV 1)

Band offset with GaAs (eV)

GaAs AlAs Ga0.5In0.5P

0.067 0.15 0.092

0.573 0.345 0.428

± 1.2 0.15

required values of the asperity height and the correlation length, for ®tting the experimental results are given in Table 2 for the two models of potential. We ®nd that the results for the Ga0.5In0.5P/GaAs system are not signi®cantly altered. On the other hand, results for the AlAs/GaAs systems cannot be explained if the roughness

Table 2 Asperity height, D, and correlation length, L‡ and L , of roughness protruding, respectively, into the barrier and the well which give agreement between calculations and experiment Sample

Eigenvalue model

Wavefunction model

Ê) D (A

Ê) L‡ (A

Ê) L (A

Ê) D (A

Ê) L‡ (A

Ê) L (A

Low mobility AlAs/GaAs

5.67

50,130

36,158

5.67

±

45,145

High mobility AlAs/GaAs Ga0.5In0.5P/GaAs

2.83 5.67

40,150 50,150

34,165 40,180

5.67 5.67

± 55,145

18,245 32,200

360

B.R. Nag, M. Das / Applied Surface Science 182 (2001) 357±360

protrudes into the barrier. It may be noted that for any value of D, the mobility varies with L such that there is a minimum value. In the case of roughness protruding into barrier in the AlAs/GaAs wells, this minimum value is much larger than the experimental value even  for D ˆ 5:67 A. The values of D and L required for roughness protruding into the well are also signi®cantly altered. In conclusion, we note that the IFRS-limited mobility in Ga0.5In0.5P/GaAs wells are approximately the same for the two models of scattering potentials. In the case AlAs/GaAs wells, the results cannot be explained by the wave function model if the roughness protrudes into the barrier. The results may be explained for roughness protruding into the well but the required values of D and L are much different. Acknowledgements B.R. Nag is indebted to Indian National Science Academy for ®nancial support under the INSA Senior Scientist Scheme and Madhumita Das is indebted to the Council of Scienti®c and Industrial Research for the award of a Senior Research Fellowship, which facilitated this study.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

S. Mori, T. Ando, J. Phys. Soc. Jpn. 48 (1980) 865. A. Gold, Phys. Rev. B 35 (1987) 723. A. Gold, Solid State Commun. 60 (1986) 5531. H. Sakaki, T. Noda, K. Hirakawa, M. Tanaka, T. Matsusue, Appl. Phys. Lett. 51 (1987) 1934. C.R. Bolognesi, H. Kroemer, J.H. English, Appl. Phys. Lett. 61 (1992) 213. J.R. Meyer, D.J. Arnold, C.A. Hoffman, F.J. Bartoli, Appl. Phys. Lett. 58 (1991) 2523. R. Gottinger, A. Gold, G. Abstreiter, G. Weilmann, Europhys. Lett. 6 (1988) 183. C.A. Hoffman, J.R. Meyer, E.R. Youngdale, F.J. Bartoli, R.H. Miles, Appl. Phys. Lett. 63 (1993) 2210. S. Elhamri, M. Ahoujja, R.S. Newrock, D.B. Most, S.T. Herbert, W.C. Mitchel, M. Razeghi, Phys. Rev. B 54 (1996) 10688. W.C. Mitchel, G.J. Brown, I. Lo, S. Elhamri, M. Ahoujja, K. Rabindran, R.S. Newrock, M. Razeghi, X. He, Appl. Phys. Lett. 65 (1994) 1578. T. Noda, M. Tanaka, H. Sakaki, Appl. Phys. Lett. 57 (1990) 165. G. Bastard, C. Deleande, Y. Guldner, M. Voos, Rev. Phys. Appl. 24 (1989) 79. T. Ando, A.B. Fowler, F. Stern, Rev. Mod. Phys. 54 (1982) 437. B.R. Nag, S. Mukhopadhyay, M. Das, J. Appl. Phys. 86 (1999) 459. B.R. Nag, S. Mukhopadhyay, Appl. Phys. Lett. 58 (1991) 1056.