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ULTRA-THIN SEMICONDUCTOR QUANTUM WELLS : POTENTIAL SHAPES AND STRAIN
EFFECTS
C. Priester, G. Allan, M. Lannoo
To cite this version:
C. Priester, G. Allan, M. Lannoo. ULTRA-THIN SEMICONDUCTOR QUANTUM WELLS : PO-
TENTIAL SHAPES AND STRAIN EFFECTS. Journal de Physique Colloques, 1987, 48 (C5), pp.C5-
203-C5-206. �10.1051/jphyscol:1987541�. �jpa-00226745�
JOURNAL DE PHYSIQUE
Colloque C5, suppl6ment
au
n O l l , Tome 4 8 , novembre 1987ULTRA-THIN SEMICONDUCTOR QUANTUM WELLS : POTENTIAL SHAPES AND STRAIN EFFECTS
C. PRIESTER, G. ALLAN and M. LANNOO
Laboratoire dlEtudes des Surfaces et Interfaces, CNRS-UA 2 5 3 , Institut Sup6rieur dlElectronique du Nord, 41, Bd Vauban, F-59046 Lille Cedex, France
RBsumB.
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La forme du potentiel dans un puits quantique trbs dtroit (de un B seize plans atomiques) est calculee dans l'approximation des liaisons fortes. Un modgle de charges ponctuelles (ajuste sur les variations de la constante dielectrique de volume en fonction de la longueur d'onde) permet d'obtenir de manihre autocoherente le potentiel r6sultant des transferts de charge au voisinage des interfaces. Dans les systbmes non contraints (GaAs/AlAs/GaAs, CdTe/Hae/CdTe), la profondeur du puits est &gale B la discontinuite de bandes de lth6t6rojonction dbs que le puits contient plus de quatre plans atomiques. De plus on a commutativit6 c'est-A-dire que le potentiel pour un systbme A/B/A est exactement l'opposi! de celui du systgme B/A/B.Par contre dans les puits contraints (constitu6s de ZnTe et CdTe ou de ZnTe et HgTe), la profondeur du puits se stabilise seulement h partir des puits de plus de huit plans, et il n'y a plus commutativit6.
Abstract.
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In the tight-binding approximation, we calculate the potential shape of an ultra-thin (one to sixteen layers) semiconductor quantum well. The potential due to charge transfers near the heterojunctions is calculated self- consistently using a point charge approximation which is fitted to reproduce the bulk dielectric constant variations with wavelength. For unstrained systems like (AlAs/GaAs/AlAs or HgTe/CdTe/HgTe) the well depth is equal to the isolated heterojunction band offset as soon as the well width is larger than four planes. Moreover, these systems are commutative (A/B/A potential is exactly opposite to the B/A/B one). But for strained wells (like ZnTe/CdTe/ZnTe, ZnTe/HgTe/ZnTe, CdTe/ZnTe/CdTe or HgTe/ZnTe/HgTe, the isolated heterojunction band offset is obtained for wells wider than8
planes and commutativity is no longer satisfied.1. Introduction.
In the effective mass approximation a semiconductor quantum well is usually treated as a potential square well whose depth is equal to the heterojunction band offset [I]. For broad wells, this is certainly a good approximation. For thin wells, a slight modification of the potential shape, depth or width affects the quantum well energy levels. We then consider ultra-thin (1 to
8
cation planes in the well) quantum wells. When the semiconductor bulk parameters are different, we assume that a semiconductor B deposited on a thick substrate made with semiconductor A, takes an in-plane parameter equal to the A one and that the interplane distance is modified according to classical elasticity. The strain effect can be separated in a variation of the charge transfers near the heterojunctions and a shift of the bands.In section 11, we briefly describe the model we use. It is an extension of previous works [2,3] for isolated heterojunctions. In order to avoid the problem of tight-binding parameters we have only considered common anion systems. The potential and the charge transfers are self-consistently calculated. In section 111, application is made to GaAl/AlAs. CdTe/HgTe, CdTe/ZnTe and HgTe/ZnTe systems. For the 11-VI compounds heterojunctions, we compare our results with recent experimental work
[4].
We notably show that the strain suppresses the band offset commutativity.Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987541
JOURNAL DE PHYSIQUE
2 . Theory
2 . 1 UNSTRAINED QUANTUM WELLS.
-
The quantum w e l l A/B/A i s made o f a few (N)[loo]
c a t i o n p l a n e s o f semiconductor B s u b s t i t u t e d t o N c o n s e c u t i v e [ l o o ] ) c a t i o n p l a n e s of an i n f i n i t e A semiconductor. A and B have a common a n i o n ( l i k e A s o r T e ) . The b u l k semiconductor band s t r u c t u r e s a r e d e s c r i b e d i n t h e t i g h t - b i n d i n g approximation u s i n g a s p 3 s * b a s i s s e t [5]. S p i n o r b i t c o u p l i n g i s a l s o i n c l u d e d . Charge t r a n s f e r s a r e t r e a t e d s e l f c o n s i s t e n t l y i n a p o i n t charge model which reproduces f a i r l y w e l l t h e b u l k d i e l e c t r i c c o n s t a n t v a r i a t i o n with wavelength. T h i s a l l o w s t o c a l c u l a t e b u l k d i p o l e l a y e r s and t o a l i g n t h e f r e e atom energy l e v e l s of t h e common anion.
Such an approach i s n e c e s s a r y a s any h e t e r o j u n c t i o n d i p o l e l a y e r i s s c r e e n e d by t h e l o n g wavelength d i e l e c t r i c c o n s t a n t 6 . A more d e t a i l e d s t u d y o f t h e s c r e e n i n g n e a r an h e t e r o j u n c t i o n w i l l be t h e s u b j e c t of a forthcoming p u b l i c a t i o n . A s both semiconductors have a common a n i o n , we avoid any problem connected t o t i g h t - b i n d i n g parameter e v a l u a t i o n n e a r t h e h e t e r o j u n c t i o n 121.
2.2 STRAINED QUANTUM WELLS. - When a few p l a n e s o f semiconductor B (whose bulk l a t t i c e p a r a m e t e r i s b ) a r e sandwidched between two i n f i n i t e s l a b s made with semiconductor A (where b u l k p a r a m e t e r i s a ) , we assume t h a t t h e l a t t i c e parameter i n t h e l a y e r p l a n e w i l l b e i n t h e whole system e q u a l t o a . We n e g l e c t any r e l a x a t i o n e f f e c t n e a r t h e h e t e r o j u n c t i o n s . A l l d i s t a n c e s bl between [100] p l a n e s i n t h e w e l l a r e e q u a l and g i v e n by c l a s s i c a l e l a s t i c i t y
[6]
T h i s v a r i a t i o n o f t h e i n t e r a t o m i c d i s t a n c e s changes t h e t i g h t - b i n d i n g p a r a m e t e r s . For t h e i n t e r a t o m i c o n e s , we have taken t h e u s u a l d'2 H a r r i s s o n law. T h i s g i v e s r e a s o n a b l e agreement f o r t h e v a r i a t i o n of gap w i t h p r e s s u r e [7]. T h i s h a s two e f f e c t s :
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t h e B semiconductor v a l e n c e band i s modified. Compared t o t h e v a l u e o f t h e band o f f s e t s , t h i s e f f e c t i s n o t always n e g l i g i b l e ( T a b l e 1 ) .The c h a r g e t r a n s f e r s n e a r t h e h e t e r o j u n c t i o n a r e a l s o modified and t h i s a f f e c t s t h e d i p o l e l a y e r .
3. R e s u l t s
3 . 1 UNSTRAINED SYSTEMS : GaAs/AlAs and HgTe/CdTe.
Ffqure 1 : Top of uaZence band acrcss the ueZS for GaAs/.i2i].s/Gcj.s a;& ;:Te/CdTe/HgTa systems. Tie numbers are the numbers of cation pZanes inside the w e i , .
Gn f i g u r e i , we have p l o t t e d t h e top of valence band f o r GaAs/AlAs/GaAs ( a ) and HgTe/CdTe/HgTe ( b ) . We s e e t h a t t h e i s o i a t e d band o f f s e t i s reached a s we have 2 B c a t i o n s p l a c e s . IT we c a l c u l a t e t h e systems AlHs/GaAs/AlAs o r CdTe/HgTe/CdTe, we g e t e x a c t l y t h e o p p o s i t e r e s u l t showing t h a t we have commutativity. The v a l e n c e band o f f s e t s we c a l c u l a t e a r e i n good agreement with experimental r e s u l t s [8,4].
3.2. STRAINED SYSTEMS : CdTe/ZnTe and HgTe/ZnTe
T a b l e 1.
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G a p edges for different semiconductors B strained to semiconductor A lattice parameter in the [100]plane (top : valence band, bottom : conduction band).Lattice parameters are respectively 6.461, 6.481 and 6.203 Angstroms for HgPe, CdTe and ZnTe.
To c a l c u l a t e t h e v a l e n c e band s h a p e , we s e p a r a t e t h e s t r a i n e f f e c t on t h e top of t h e v a l e n c e band ( T a b l e 1 ) and on t h e d i p o l e l a y e r . For i n s t a n c e i f we consider CdTe/ZnTe/CdTe, we g e t a d i p o l e l a y e r e q u a l t o
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0.67 eV t o which we must add a s h i f t of t h e v a l e n c e band (Table 1) e q u a l t o + 0.50 eV. T h i s g i v e s a valence band o f f s e t e q u a l t o-
.17 eV. The same q u a n t i t i e s f o r ZnTe/CdTe/ZnTe a r e + 0.57 eV and+ 0.37 eV which g i v e s a v a l e n c e band o f f s e t of .94 eV. I f we assume no s t r a i n a t t h e ZnTe/CdTe h e t e r o j u n c t i o n , w e g e t a d i p o l e l a y e r e q u a l
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0 . 4 1 eV f o r CdTe/ZnTe/CdTe and + 0 . 4 1 eV f o r ZnTe/CdTe/ZnTe. Commutativity i s s a t i s f i e d . But a s s t r a i n s a r e o p p o s i t e i n t h e w e l l , t h e e f f e c t s on t h e d i p o l e l a y e r a r e a l s o o p p o s i t e g i v i n g r i s e t o almost e q u a l a b s o l u t e v a l u e s of s t r a i n e d d i p o l e l a y e r s . As t h e o r i g i n h a s always been taken o u t s i d e t h e w e l l , we must compare t h e a b s o l u t e v a l u e s . The v a l e n c e band o f f s e t s a r e q u i t e d i f f e r e n t even i f t h e d i p o l e l a y e r s a r e v e r y c l o s e . On f i g u r e 2 , t h e t o p o f t h e v a l e n c e band i s p l o t t e d a c r o s s t h e w e l l . The non commutativity i s c l e a r . Not o n l y t h e amplitudes a r e d i f f e r e n t b u t a l s o t h e w e l l shape.A B HsTe
CdTe
ZnTe
Figure 2 : Top of valence band across the well for CdTe/ZnTe/CdTe and ZnTe/CdTe/ZnTe systems. T!:z n.~mb.zrs are the nt;mbe-s op cation planes inside the weZZ.
CdTe -0.01
+o.oi
0 . 0 0
1.59
0.50 2 . 1 1 HgTe 0.00 0.00 0 . 0 1 1 . 6 0 0.48 2.17
ZnTe -0.14 + 0.17
0.37 1.85 2.56
C5-206 JOURNAL DE PHYSIQUE
Measurements o f t h e c o r e l e v e l s near an h e t e r o j u n c t i o n g i v e d i r e c t a c c e s s t o t h e d i p o l e l a y e r o r i t s v a r i a t i o n s . Recent measurements [4] o f t h e c a t i o n c o r e l e v e l energy f o r t h e CdTe/ZnTe/CdTe and ZnTe/CdTe/ZnTe systems g i v e d i p o l e l a y e r v a r i a t i o n e q u a l t o 0.17 eV which must be compared t o c a l c u l a t e d v a l u e e q u a l t o 0.10 eV. The same q u a n t i t i e s f o r HgTe/ZnTe a r e r e s p e c t i v e l y 0.09 e V and . I 7 eV. The agreement
( s i g n and amplitude) between t h e o r y and experiment i s q u i t e good.
I n c o n c l u s i o n , we have p o i n t e d o u t t h e i n f l u e n c e o f s t r a i n on t h e p o t e n t i a l shape and v a l e n c e band o f f s e t s . I f t h e a b s o l u t e v a l u e s of t h e d i p o l e l a y e r a r e n o t very d i f f e r e n t , t h e s t r a i n g e n e r a l l y s h i f t s upwards t h e t o p o f t h e v a l e n c e band (except f o r HgTe) b r e a k i n g t h e commutativity i n m u l t i p l e h e t e r o j u n c t i o n systems.
References
[ I ] See f o r example M. ALTARELLI i n "Heterojunctions and Semiconductor S u p e r l a t t i c e , Proceedings o f t h e Winter School. Les Houches. March 1985, G. Allan, G. Bastard, N. Boccara. M. Lannoo and M. Voos e d . , S p r i n g e r Verlag, p.12
[2] B. HAUSSY, C. PRIESTER. G. ALLAN and M. LANNOO, Phys. Rev. ( p u b l i c a t i o n scheduled f o r 1 5 J u l y 1987).
[3] A. MUNOZ, J . C . DURAN and F. FLORES, Phys. Rev.
m. . 1987.
[4] TRAN MINH DUC. C. HSU and J.P. FAURIE, Phys. Rev. L e t t .
3.
1129 (1987) concerns t h e (111) system, b u t an unpublished work by t h e s e a u t h o r s concerns t h e (100) system.153 G a A s , A l A s . ZnTe b u l k parameters a r e c o r r e c t e d v a l u e s from P. VOGL, M.P.
HJALMARSON and J . D . DOW, J. Chem. S o l i d s ,
44,
365 (1983). For CdTe and HgTe, we u s e A. Kobayashi, O.F. Sankey and J . D . Dow, Phys. Rev.g ,
6367 (1982).i n c l u d i n g s p i n o r b i t coupling.
[6] J.Y. MARZIN, i b i d . r e f . [I], p. 161.
[7] G. MARTINEZ, i n Handbook on Semiconductors, V.2. M. Balkanski ed., North Holland (1980). p. 181.
[8] Recent measurements o f G a A s / A l A s o f f s e t s g i v e .49 eV [ W . I . WANG, T.S. KUAN, E.E.
MENDEZ and L. ESAKI, Phys. Rev.
m,
6890 (1985)1.
.45 e V [M. HEIBLUN. M . I . NATHAN and M. EIZENBERG, Appl. Phys. L e t t . , Q, 503 (1985)], .55 eV [J. BATEY and S.L. WRIGHT, J . Appl. Phys.,3,
200 (1986)] and .54 eV [P. DAWSON. B.A.WILSON, C.W. TU and R.C. MILLER, Appl. Phys. L e t t . ,
