Momentum mixing enhancement of the conduction band nonparabolicity in GaAsGa1−xAlxAs, GaAsGaAs1−xPx and SiSi1−xGex superlattices

Superlattices and Microstructures, Vo/. 6, No. 2, 1989

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MIXING E~'IANCKM]~T OF TH]E CONDUCTION BAND NONPARA]IOLICITY IN

C~s-C~z_=Al~ , c.~A=-C.m~=_~P=AND m-ms_=G% s ~ l " r l c ~ , L,D.L. B r o w n . R.J. Turton and M. Jaros Department of Theoretical Physics, The University, Newcastle Upon Tyne Tyne and Wear, NEt 7RU United Kingdom

( Received 8 August , 1988 )

We p r e s e n t pseudopotentlal calculations of t h e nonlmrabolicity at t h e minimum of t h e lowest superlattlce conduction mlniband. The mechanisms affecting t h e band curvature in three superlattice systems, GaAs-Gal_xAlxAS, GaAs-GaAs1_xP x and Si-Sia_xGe x are compared. Significant e n h a n c e m e n t of t h e nonparabolicity over bulk materials is observed in each case.

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The study o f materials with nonparabolic bands is o f particular interest because o f the applications t o nonlinear optical processes. We compare and c o n t r a s t t h e nonparaboltclty of t h e l o w e s t conduction miniband in t h r e e s y s t e m s , GaAs-Ga~_x AI As, GaAs-GaAsl_xPx snd $i-Sll_xGex,the latter; two I~eing strained layer superlattices. The band curvature is affected by transitions between the zone-folded minima, an effect first proposed by Tsu and Esaki I ) in 1970. Recent pseudopotential calculations 2.3) have revealed t h a t t h e superlattice microscopic potential and t h e s t r a i n combine bulk s t a t e s o f different m o m e n t a In t h e superlatUce states. This m o m e n t u m mixing enhances t h e transition probability and t h e r e fore t h e mlnlband nonparabollcity.

2. Method ~ CalcuMtion We use a pseudopotential scheme in which the super.lattice eigenl:unctions ~_ are g e n e r a t e d in t e r m s o r a iineer combination o f t h e eigenfunctions ~rdc= of a suitably chosen bulk Hamiltonian. Thus ~b =nJ~ Anks~mk= where n is the bulk band Index, k is the wavevector and s is t h e spin variable. The Si-Sil.xGe x calculation does n o t Incorporate the e f f e c t of s p i n - o r b i t coupling, t h e r e f o r e the spin variable is dropped. The eigenfunctions ~ a k in the G a A s - G a l _ x A l x A s calculation are chosen t o be solutions o f t h e bulk GaAs Hamfltonisn. In t h e strained superlattices t h e bulk eigenfunctions are calculated for a material representing an average alloy of t h e superlattice, 4 ) u s u m e d t o be t h e b u f f e r layer on which t h e superlattice is grown. In this system t h e strain Is evenly t a k e n up by t h e two c o n s t i t u e n t layers of t h e superlattice.

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The p r o c e s s e s causing the band nonlmrabolicity are t h e virtual transitions between t h e superlattice minlbands. A second order calculation s h o w s t h a t t h e virtual transitions between t h e conduction minib a n d s t e n d t o depress t h e conduction ground istate energy, whilst transitions across t h e superlattice band gap raise the energy. The magnitude,, o f t h e noBparabolicity is determined by calculating t h e f o u r t h derivative o f t h e energy with respect t o w a v e v e c t o r , ( d t E / d k t ) , which is p r o p o ~ g n a l t o t h e t h i r d o r d e r nonlinear susceptibility XI ° " . ( Note that in all of t h e s e s t r u c t u r e s t h e symmetry rules o u t large values o f d 3 E / d k °. ) T h e ' f o u r t h (iertvntlve is evaluated u s i n g Kane's k • p p e r t u r b a t i o n theory, s) The explicit expression a t t h e Brillouin zone c e n t r e being 8 t v e n in equation (2) of [~eference 6. The t e r m s in tim expression are proporUomfl t o t h e p r o d u c t of fottr matrix elements and Inversely proportional t o t h e product of t h r e e energies. The energies are measured from the g r o u n d s t a t e conduction band, t h e r ~ o r e t h e s t a t e s n e a r e s t In e n e q ~ t o t h e ground s t a t e t e n d to m o s t affect t h e nonpsrabollctty.

3. ilmuits Ga~s-Ga1~x AI x As Bulk GaAa has a signffic~tl~, n o - - t i c c o n a u c t i o n b a n d due to t h e s t r o n g direct t r a n s i t i o n across t h e bmtd gap.. In t h e GaAs-Ga/_xAlxAS Superlattice, trLnsitiona oetween t h e l o w e s t auper[attice minlbands also affect t h e band curvntuvei The

conduction band con,t~buuon to the fourth dsrlv=Uve m c m a e s t r m ~ l t i o n s oecween t h e F m m l m a m a d t h e zone folded s t a t e s o r d i n a t i n g from t h e bulk X minim& This transition probability may be e ~ t o be very small. However t h e superlattice micro-

© 1989 Academic Press Limited

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Superlattices and Microstructures, Vol. 6, No. 2, 1989 V- . . . . . . . . . . . . . . . . . .

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Figure I - The variation of the ground state conduction band no_nparaboliclty in a GaAs ( SO ~ Ga1_xAlxAS ( 22 X ) superlattlce as a function }of aluminum fraction x. The nonparabollcity is expressed in units of the bulk GaAs conduction band nonparabolicity evaluated at the centre of the bulk Brillouin zone. The upper curve indicates the nonparabollcity calculated using the full pseudopotential calculation, whilst the lower curve represents values obtained from a Kronlg-Penney model. The difference illustrates the importance of m o m e n t u m mixing effects, which can only be Incorporated in the full calculation.

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Figure 2 -- The variation of the ground state conduction band nonparabollcity with P mole fraction x in a GaAs ( 20 ~ ) - GaAs1-~Px ( 20 ~ ) superlattice. As In figure I, the nonparabollcity is expressed in terms of the bulk GaAs conduction band nonparabollclty.

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scopic potential mixes bulk states of different m o m e n t a into the superlattice eigenfunctions so that the transition probability becomes finite.Consequently, transitions between the superlattice mlnibands dominate the mlniband curvature increasing the nonparaboliclty by nearly an order of magnitude over bulk GaAs 6). Comparison with a simpler KronigPenney model, in which m o m e n t u m mixing is absent, illustrates the importance of m o m e n t u m mixing on the nonparabollclty. (Figure I ).

GaAs-GaAs#-x Px The GaAs-GaAs1_ x Px system is a strained layer superlattice, the strain increasIng with the fraction of P contained within the GaAsl_xP x layer. As in GaAs-Ga1_ x Al× As the nonparabolicity of the ground superlattice conduction state is principally affected by t r a n s i t i o n s b e t w e e n the r m i n i m u m and t h e folded X minimum. The m o m e n t u m mixing and t h e r e f o r e

the transition probability between these states increases as the strain is increased, ie for increasing x. until x r e a c h e s a value o~" ,~O.S ( Figure 2 ) . Beyond this value a l t h o u g h t h e s t r a i n increases the m o m e n t u m mixing, the e f f e c t o n the n o n p a r a b o l i c i t y is reduced as the c o n d u c t i o n s t a t e s b e c o m e f u r t h e r s e p a r a t e d in energy, in addition t h e c o n t r i b u t i o n f r o m the valence band b e c o m e s l a r g e r which a c t s a g a i n s t t h e e f f e c t s f r o m t h e h i g h e r c o n d u c t i o n s t a t e s , y) The n o n p a r a b o i i c i t y can be f u r t h e r increased by extending t h e super~attice period ( Figure 3 ), since this d e c r e a s e s t h e energy s e p a r a t i o n o f the l o w e r m o s t conduction states.

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Figure 3 -- The conduction band nonparabollclty at the zone centre for a GaAs - GaAso.sPo. s superlattice as a function of the superlattice period. The constituent layers are assumed to be of equal width, with the strain distributed evenly in the two materials. The nonparabollcity is measured in units of the bulk GaAs conduction band nonparabollcity.

Si-SiI_xGe × In b u l k s i l i c o n , the i n d i r e c t t r a n s i t i o n across the

fundamental gap is very weak, consequently the conduction band is virtually parabolic in the region of the minimum. In the Si-Si1_xGe x superlattice, the combined effects of zone-folding and m o m e n t u m mixing enhance the oscillator strength of the transition across the band gap to within two orders of magnitude of that associated with a direct gap material 2). Comparatively strong transitions also arise between the lowermost conduction minibands. These virtual transitions dominate the nonparabolicity due to the small separation in energy of the states

Superlattices and Microstructures, Vol. 6, No. 2, 1989 f r o m the conduction g r o u n d state, producing a value of d 4 E / d k ~ c o m p a r a b l e with t h a t o f bulk GaAs. Table 1 The variation o f the g r o u n d s t a t e c o n d u c t i o n band nonparabolicity for Si-Sil, x Ge x superlattice evaluated at the band m i n i m u m w i t h s u p e r l a t t i c e period is s h o w n in the t a b l e below. The values are q u o t e d in u n i t s of the bulk GaAs c o n d u c t i o n band nonparabolicitv. period (,~) d4E/ dk ~ ll.S O.OS 22.0 0.16 27.5 0.63 Since the relevant s t a t e s originate f r o m the s a m e region of the bulk Brillouin zone. i.e. near the I minima, m o m e n t u m mixing does n o t significantly e n h a n c e the t r a n s i t i o n r a t e s b e t w e e n t h e s e s t a t e s . The f o u r t h derivative is t h e r e f o r e s t r o n g l y d e p e n d e n t on the energy s e p a r a t i o n of the l o w e r m o s t c o n d u c t i o n s t a t e s , being inversely p r o p o r t i o n a l to t h e cube of the energy difference. This can lead to a variation in the f o u r t h derivative o f at least t h r e e o r d e r s o f magnitude across the s u p e r l a t t i c e Brillouin zone. The f o u r t h derivative at a given w a v e v e c t o r can be m a x imised by s u i t a b l e t u n i n g o f the energy levels.

4. Discussion The t r a n s i t i o n s which a f f e c t the band c u r v a t u r e of the g r o u n d s t a t e c o n d u c t i o n miniband are f r o m higher c o n d u c t i o n s t a t e s , and f r o m t h e valence s t a t e s . The higher c o n d u c t i o n s t a t e s are a r e s u l t of the z o n e - f o l d i n g in the s u p e r l a t t i c e . The m a g n i t u d e o f the nonparabolicity increases as the m a t r i x e l e m e n t s b e t w e e n the s t a t e s increase. and as the energy s e p a r a t i o n o f the s t a t e s f r o m the g r o u n d s t a t e decreases. The m a g n i t u d e s of the m a t r i x e l e m e n t s b e t w e e n the s t a t e s are d e t e r m i n e d by the need f o r c o n s e r v a t i o n o f m o m e n t u m , being l a r g e s t when the t w o s t a t e s have a p p r o x i m a t e l y the s a m e m o m e n t u m . The t r a n s i t i o n s b e t w e e n the l o w e r m o s t conduction s t a t e s in the GaAs s y s t e m s , and a c r o s s the gap in the Si-Si~_xGe x s u p e r l a t t i c e , are b e t w e e n s t a t e s derived f r o m U and f r o m X. i.e, the s t a t e s have very different m o m e n t a . H o w e v e r the m o m e n t u m mixing produced by the s u p e r l a t t i c e m i c r o s c o p i c potential, and additionally by the s t r a i n in t w o o f t h e s y s t e m s , e n h a n c e s t h e s e t r a n s i t i o n probabilities considerably. For the s y s t e m s c o n s i d e r ed here. the energy s e p a r a t i o n o f the g r o u n d

165 c o n d u c t i o n s t a t e to the higher s t a t e s is much s m a l l e r t h a n t h a t to the u p p e r m o s t valence s t a t e s . O f the t r a n s i t i o n s affecting the n o n p a r a b o l i c i t y in the S i - S i l _ x G e x s u p e r l a t t i c e s t h o s e b e t w e e n the l o w e r m o s t conduction s t a t e s d o m i n a t e since the energies, as m e a s u r e d f r o m the g r o u n d s t a t e , are small and the matrix e l e m e n t s involved are larger t h a n the enhanced t r a n s i t i o n s a c r o s s the s u p e r l a t t i c e gap. in the GaAs s y s t e m s the transition across the gap is s t r o n g e r , w h i l s t the t r a n s i t i o n s b e t w e e n the c o n d u c t i o n s t a t e s are weaker. However. in general the nonparabolicity is still d o m i n a t e d by t r a n s i t i o n s within the c o n d u c t i o n band. due to the small energy separations. The d o m i n a n t t r a n s i t i o n s in the Si-Si~_xGe ~. s u p e r l a t t i c e s are b e t w e e n states with similar m o m e n t a . The m o m e n t u m mixing e f f e c t s are small. and the nonparabolicity depends m o s t s t r o n g l y on t h e c h a n g e s in the energy levels. The GaAs s y s t e m s are more c o m p l e x since the t r a n s i t i o n probability b e t w e e n the c o n d u c t i o n s t a t e s depends s t r o n g l y on the degree of m o m e n t u m mixing. To obtain the m a x i m u m nonparabolicity, it is t h e r e f o r e n e c e s s a r y t o s i m u l t a n e o u s l y o p t i m i s e the m o m e n t u m mixing and the energy s e p a r a t i o n of the c o n d u c t i o n energy levels, A c k n o w l e d g e m e n t s - We would like to t h a n k SERC ( U, K. ) and British T e l e c o m Research Laboratories f o r financial s u p p o r t .

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