Two-dimensional donor-bound excitons in the extreme magnetic quantum limit

Physics Letters A 173 (1993) 311-316 North-Holland

PHYSICS LETTERS A

Two-dimensional donor-bound excitons in the extreme magnetic quantum limit Alexander B. Dzyubenko ResearchCentrefor TechnologicalLasers,AcademyofSciencesofRussia, 142092 Troitsk,MoscowDistrict,RussianFederation Received 23 November 1992; accepted for publication 18 December 1992 Communicatedby V.M. Agranovich

Two-dimensional (2D) impurity-boundexciton complexesare studied in the extremequantum limit of a magneticfield B. For a 2D magnetoexcitonbound to the ionized donor (D +, X ) the localized spectra consist of (i) the discrete low-lyingbound states below the free magnetoexciton (MX) band, (ii) the resonant states with finite widths within the MX band, and (iii) the excited discrete states above the MX band. Stable states of a magnetoexeitonbound to the neutral donor (D°, X) and the magnetobiexciton bound to the ionized donor (D +, BX) with electrons in the spin-singlet state are found. Optical transitions associated with 2D impurity-boundexeiton complexesin a strong magneticfield B are discussed.

Far-infrared magneto-optics of shallow-impurity states in semiconductor quantum wells (QWs) has been widely used in recent years to probe in detail the dependence of energy levels on the QW width, on the position of the impurity and to study the magnetopolaron effects (see e.g. ref. [ 1 ] ). Relatively few papers have been devoted, however, to the magneto-optics of the impurity-bound exciton complexes in I I I - V semiconductor QWs (see ref. [ 2], and references therein). Most of the experimental measurements (see ref. [3 ], and references therein) and the theoretical work [ 4-6 ] deal with the quasi-2D excitons bound to the ionized and neutral donors in the absence of an external field B. In this paper (which is a continuation of refs. [ 7,8 ] ), we report on the detailed study of the spectra of strictly two-dimensional (2D) magnetoexcitons bound to the ionized (D+, X) and neutral shallow donor (D °, X) and on the search for stable donor-bound multiexciton complexes containing up to three e-h pairs in the extreme quantum limit of a magnetic field B (electrons and holes are confined to their spin-split zero Landau levels 0L 07, while higher Landau levels are neglected, which is valid when IB=(~c/eB)l/2<
eh2/me
~ m2--ml

~£~~h)(rel,re2;rh)=

A~(m~;m2)O~)t(re) 0 (mh2)(rh) ,

(1)

~

(2)

=g

A~r(n,m;l)fb~)(r)O~me)(R)O~h)(rh),

I--m--n=M

where 0 ~ ) ( r ) = ¢ ~ ) * ( r ) are the wave functions corresponding to the zero Landau level, R = (re, + re2)/v/2, r= (tel -re2)/v/2 and n=2p for the singlet S~=O and n=2p+ 1 for the triplet S~= 1 electron states; see refs. ElsevierSciencePublishers B.V.

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[7,8] for more details. For the complex (D +, 3e-h) (or, equivalently, ( D - , X ) ) the wave function is deterflained as

~;h)(r~l,r~2, r~3;rh)=

~,

AM(i,k,n;t) (r~l,r~2, r~31i, k,n)s,O}h)(rh),

(3)

l--i--k--2n=M

where Ii, k, n )s, is the state of three electrons in the zero Landau levels 0J,, Ot with total spin S~ = 3 or S~= ½ [9] (see also ref. [ 10], and references therein). The wave functions of the (D °, BX) and (D ÷, 3e-3h) complexes, ~ l~S, 3~-2h) Sh and t£t(3e-3h) "~S~S~ , are constructed analogously. The expansion coefficients A~, are to be determined by solving the secular equation involving the Hamiltonian of the Coulomb interactions which (in the general case with N¢ electrons and Nh holes) has the form Ne

e2

Nh

i
e2

Ne Nh

t~lrk--rt[

C2

s s ~lri--rkl "

N= C2

t

z -~r~ +-. k •

C2

(4)

~rk

In a magnetic field B the quantization of angular momenta is directly related with the quantization of orbitals in real space. Hence, the wave functions from the constructed set are localized in different parts of r-space, thus providing a systematic and convergent scheme for studying the impurity-bound states [ 7,8 ]. In the present calculations from 60 to 150 orbitals are taken into account. We estimate the achieved absolute accuracy in determining the interaction energies to be of the order and ranging from 10-6Eo for (D +, X) (see Eo in eq. (5) below) to ~ 10-2E 0 for (D +, 3e-3h). Notice also that the angular momentum projection M turns out to be directly related with the distributions of electrons and holes (which revolve in opposite directions). For, e.g., the (D +, X ) complex, one has (r 2 - r ~2) M = 2 M l 2 [7,8]; a general trend for e-h complexes is that for M > 0 holes are more in the outer orbitals. For the (D +, X) complex for each M > 0 there is one discrete low-lying impurity-localized state below the exciton band (see figs. 1 and 2). The binding ener y~_~b (M) is a powerqaw decreasing function of M with the largest value Eb(l ) = 0.1189E0 ~ B 1/2; here Eo = x/re/2 e2/EIB is the binding energy of a 2D MX of zero momentum which strictly coincides with the binding energy o f a 2D magnetoimpurity ground state D O [ 11 ] (see eq. (5) below for V~~)). 0.05 - - ~ . ,

0.16

, .........

, ...................

2

'~

(D+,X) M = I : DISCRETE LOW-LYINC

°_ooo

LEVEL

CO 0.12 LJ CO Z bJ

0.08

- ......

>-

uS - 0 . 0 5 Z Lg

electron densTty hole density

o <( r~

<:~ 0.04 0

(D+,x) M=I +++++ M=3 .-~m-H M = 6 o-e-oe-~ M = 1 5 xx××x

,] ,~ '

0.15

~

O:~

[rL

".

0.00

-0.20 0.0

2,0 DISTANCE

4.0 FROM

6.0 IMPURITY

8.0 (~a)

Fig. 1. Electron and hole probability densities, pc(r), ph(r) (in units of 1~2 =eB/hc) of the ground (D +, X) state with M = 1 (see also table 1 ).

312

/;t

-OlO / :

........ 0.0

, ............................ 2.0 4.0 6.0

DISTANCE

FROM

IMPURITY

8.0 (F,B)

Fig. 2. Charge densities e[ph(r) --pc(r) ] for four impurity-bound discrete low-lying states of (D +, X) with M = 1, 3, 6 and 15. Their binding energies (in units ofEo = x / ~ e2/ds) are given by, respectively, Eb=0.119, 0.069, 0.021 and 0.004.

Volume 173, number 3

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Due to a localizing effect of a magnetic field B, there appear two distinct features of the MX spectra in the presence of D+: ( 1 ) For M~> 5 there are resonant states with finite widths which lie within the free MX band. Physically, these states correspond to virtual states of a hole bound to the excited neutral donor (D*, h). The energies of these states are determined (see also refs. [ 7,8 ] ) by the matrix elements of the electron-impurity interactions in the zero Landau level, v,.,:, =-

(2m- 1 )~ Eo --mm.ti

'

Eo=

/~- e2 ~ B 1/2

(5)

(cf. the discussion of the 2D D - spectra in the zero Landau levels in refs. [ 10,12 ] ). Figure 3 shows the lowestlying resonances corresponding to a D* in the first excited state near (and below due to e - h interactions) the energy V p ) = - 0 . 5 E o . For M = 6 the resonance is rather broad and in the obtained (D +, X ) spectra which fill in the free MX band there are several states (shown are only two) containing the localized part; the difference in their energies BE= 0.05Eo is a rough measure of the resonance width. With increasing M the resonance becomes narrower and in the limit M>> 1 it corresponds to an electron in the first orbital m e = 1 (with energy V p ) = - 0 . S E o ) weakly interacting with a hole which is at a large distance from the impurity (in the orbital mh = M + 1 ). For large M there also appear resonances corresponding to higher D* states. The existence of the resonant (D +, X ) states has been also confirmed by the direct solving [ 13 ] of the scattering problem of a 2D MX by the ionized donor. (2) For M < 0 (when a hole is closer to the donor ion D ÷ than an electron) there are excited discrete impurity-bound states which lie above the MX band (the latter has a finite width Eo [ 11 ] ). The total charge densities for several such states are shown in fig. 4; note that for M = - 5 there are two discrete excited levels while shown is only the highest one. In the limit M - - , - oo these excited levels correspond to an electron at infinity and a hole bound in a strong field B to a positive donor in the states with different magnetic quantum numbers mh = 0, 1.... and with positive energies V ~ ) = - F ~") (see eq. (5) and ref. [ 11 ] ). The interaction of the 2D s-MXs (angular m o m e n t u m projection M = 0) with an axially symmetric external field is absent. This is due to the fact that in each m channel involved in expansion ( 1 ) the electron- and holeimpurity interactions are cancelled: V ~ ) = - V~~). According to the matrix element for the optical transition 0.20

0.050 &'Lm '-L"~-"O.040 dch

~

(D+,X): RESONANT STATES

I~ 0J

I \ [ \ / \ / /x, \

I--

U') 0.0,30 LdZ n >- 0.020 F-_

M=IO - - - M=8 ,a-,a-~a,~,M=6 ooeee M : 6 ---

E=-0.5.3 E=-0.54 E=-0.57 E=-0.52

nq .< m 0.010 0 rY o.ooo

#, ..............

0.0

2.0

DISTANCE

4.0 FROM

......

6.0

IMPURITY

8.0 (gB)

Fig. 3. Electron probability densities p,(r) of the localized resonant states of (D +, X) with different values of M lying within the free MX band.

0.15

l

,r

. . . . . . .

I

. . . . . . . . .

I

. . . . . . . . .

~ . . . . . . . . .

(D+,X): M
~

DISCRETE

EXCITED

LEVELS

>-

O.lO z La_l a td 0.05 LD Od
~'~, ~, ,~,,

..... M = - I E=0.042 ~''~'~'~ M = - 3 E=0.361 o-ee-o-eM = - 5 E=0.497

'~-,z--~-~-.~s

0.0

. . . . . . . . .

i

. . . . . . . . .

2.0

;

i

4.0

. . . . . . . . .

i

.~ =~ - -

. . . . . . . . .

6.0

8.0

DISTANCE FROM IMPURITY (gB) Fig. 4. Charge densities and the energies of the discrete excited (D +, X) states with M = - 1, - 3 , - 5 which lie above the free MX band (cf. fig. 2).

313

Volume 173, number 3

8 February 1993

2

I t .

W~evJ

PHYSICS LETTERS A

dr ~tl~-h)(r; r) ~ g . o

(6)

obviously, it is s-MXs which are optically active; here Pcv =P~v (k = 0) is the m o m e n t u m interband matrix element. More precisely, only the lowest s-MX state o f the zero magnetic (see ref. [11 ]) m o m e n t u m e-h ~v~=~.lkl =o (or, equivalently, ~ - h ) ) is optically active ~1 For this 2D MX state there is no interaction with arbitrary electrostatic external fields [7] (not only the axially symmetric fields as for all s-MXs) ~2. When (i) a difference between the wave functions of the motion of electrons perpendicular to a QW, ~e(Z), and that of holes, ~h(Z), and (ii) the admixture of higher Landau levels are taken into account, there appears a weak interaction o f s-MXs with D ÷. The former effect in the mth channel is described by (compare with eqs. (6), (7) o f ref. [ 8 ] )

~-

~-

-

f

d2q 2 h ez ( 2 n ) : ~q L ' ( ½ q : l ~ ) e x p ( - ½ q : 1 2 ) [ F e ( q ; z ° ) - F n ( q ; z ° ) ] '

(7)

where Lm is the Laguerre polynomial and F~
dzexp(-qlz-zol)

~Z~n)(Z)

(8)

are the form factors corresponding to the lowest subbands, and Zo is the position o f the impurity in a QW. Thus, s-(D +, X) may have rather small binding energy (which should strongly depend on the position o f the impurity within a Q W ) and, hence (see ref. [20] ), giant oscillator strengths for optical transitions. In a situation when the electron and hole effective masses and the well depths for electrons and holes do not differ much, it m a y be even expected that an increase o f B (while deepening higher-momentum (D +, X ) states) may cause unbinding o f s-(D +, X ) which may be observed in the optical spectra. For (D °, X) the only found stable state is the spin-singlet with M = 1 and very low binding energy E b = 0.005Eo so that the Haynes rule is inapplicable. It is worth noting here that for a (D °, X) complex (presumably due to the total electroneutrality) convergence turns out to be more slow than for a (D +, BX) complex. In our previous calculations [7,8] involving a rather limited set of orbitals, the existence o f the stable bound singlet (D °, X) state in the zero Landau levels (which is almost obvious on physical grounds) has not been established. Here again, as in the case o f the s-(D +, X) complex we expect that the inclusion of quasi-2D effects and the admixture o f higher Landau levels is o f qualitative importance. In particular, it should be expected that a stable b o u n d s-(D °, X) state in intermediate fields is formed. The stable states o f a 2D magneto-biexciton b o u n d to the ionized donor (D +, BX) with electrons in the spin-singlet state are also found (see table 1 ). All other states of these complexes, as well as of the studied 2D multiexciton complexes (D +, 3 e - h ) , (D °, BX) and (D +, 3e-3h) in all possible spin states turn out to be unstable against a removing o f a 2D free MX. Probability and charge densities depicted in fig. 5 show that the sizes of the stable impurity-bound exciton states in a strong magnetic field do not change appreciably with increasing the number of particles in a complex (compare also figs. 1 and 2). #~ For the 2D MX with k = 0 the expansion coefficients from eq. ( 1) are given by Ak=o(m; m) = N 6-1/2 where No = S/2~12 is the macroscopic degeneracy of Landau levels, and S the area of the system. Formally, this follows from the fact that in the total angularmomentum representation the infinite Hamiltonian matrix of e-h interactions ( see eqs. (6), ( 7 ) of ref. [ 8 ] ) for M= 0 is stochastic (see e.g. ref. [ 14 ] ), i.e. according to the sum rule ~ ~-o U~'~m'= Eo the sum of the elements in each row is constant. Note that due to the exponential decay of U,~' with Irn-re'l, for the delocalized k=0 ground MX state the accuracy in determining its binding energy (i.e. the lower MX band edge) by a diagonalization of a finite 100× 100 Hamiltonian matrix is 0.7%. ~2 This property is a manifestation of the ideal character of 2D free MXs of zero momentum forming a gas of noninteracting composite particles (see refs. [ 15-17 ] ). The indications of this very unusual property have been obtained in the recent magneto-optical experiments [ 18,19]. 314

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Table 1 Energies (in units of Eo) and quantum numbers of the stable impurity-bound 2D exciton complexes in the zero Landau levels. For a comparison the results for the ground states of a 2D free MX, a neutral donor D Oand the negative donor center D- in the zero Landau levels are also shown. Complex

Ref.

Spin state

MX DO D-

[11] [11]

(D +, X) (D °, X ) (D +, BX)

[7,8] this work [7,8]

[9,10,12,21 ]

S®ffi0 S,= 1 Se= 0 Sc=0, Sh= 1 S,=0, Sh=0

0.16

Angular momentum M,

Interaction energy (Eo)

Binding energy (Eo)

0

-1

1

0

-1

1

- 1.2929 - 1.1465 - 1.1189 - 2.005 --2.174 --2.188

0.2929 0.1465 0.I 189 0.005 0.055 0.069

0 - 1 1 1 3 2

.,---,, 0.05 l

--(D°,X)

.........

i .........

i .........

i .........

S,,=O M = I 0.00

b-oo

(./3 0.12 Ld

/

7

Z L.d cm 0.08 >-

E3 - 0 . 1 0 i,i (_3 r~ <~ ,' n"~ ,'/ O -0.2O

<~ 0 . 0 4 CO 0 r'f EL

0.00

..... ---

) S,=O M = I " (D+,BX') S.=O Sh=O M = 2 (D+:BX) Se=O Sh=l M = 3

_] .<

o 0.0

2.0 DISTANCE

4.0 FROM

6.0

8.0

IMPURITY

(ga)

(b) -0.30

0.0

2.0 DISTANCE

4,0 FROM

6.O IMPURITY

8.0 (~)

Fig. 5. Electron and hole probability densities per one particle (a) and the total charge densities (b) of the stable bound states in 05, 05 Landau levels of the 2D magnetoexeiton bound to the neutral donor (D °, X) and the magneto-biexciton bound to the ionized donor (D ÷, BX). C o n s i d e r now the optical selection rules in a strong field B for a r e c o m b i n a t i o n o f an e - h pair from the ( D °, X ) complex with the wave function ~U~-h) (r.~, r,2; rh) (the spin q u a n t u m n u m b e r s are o m i t t e d ) . The matrix element for the optical transition to the final electron state ~f(r) has the form 2

(9) T h e conservation o f the angular m o m e n t u m in the transition m f - M implies that only the optical transitions involving the s- ( D °, X ) (with M - - 0 ) are strong in high magnetic fields. Indeed, from the initial ( D °, X ) state with positive M > 0 (these states are expected to be the most strongly b o u n d in a quasi-2D situation, while the states with M < 0 are expected to be u n b o u n d ) , the achievable final electron states are only in the M t h or higher L a n d a u levels. In a strong magnetic field, when the a d m i x t u r e o f these L a n d a u levels to ~ - h ) ( r e ~ , re2; rh) is small, the r e c o m b i n a t i o n spectra o f the ( D °, X ) with M > 0 should consist o f a series o f weak lines associated 315

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with the optical t r a n s i t i o n s ( i n which a n electron is excited to the Y t h L a n d a u level Y = M , M + 1 .... ) whose i n t e n s i t i e s fall off as (Eo/Xhcoce) 2 ~ B - 1, where co~ = eB/rnec. T h e results o b t a i n e d are q u a l i t a t i v e l y applicable to 8 - d o p e d s t r a i n e d - l a y e r Q W s (e.g. I n G a A s / G a A s Q W s ) with a n o n d e g e n e r a t e v a l e n c e b a n d a n d to heavy-hole exciton complexes in G a A s / G a A I A s Q W s in a strong q u a n t i z i n g p e r p e n d i c u l a r field B. F o r a q u a n t i t a t i v e d e s c r i p t i o n o f the e x p e r i m e n t a l situation, (i) the m i x i n g b e t w e e n L a n d a u levels (especially for m o r e h e a v i e r holes), (ii) the m o t i o n p e r p e n d i c u l a r to a Q W a n d (iii) ( i f p r e s e n t ) the c o m p l e x character o f the v a l e n c e b a n d are to be i n c l u d e d into the p r e s e n t c o n s i d e r a t i o n . T h i s work is n o w i n progress. Note added. After this work h a d b e e n completed, we b e c a m e aware o f the recent e x p e r i m e n t a l studies o f B u h m a n n et al. [22 ] o f the r e c o m b i n a t i o n o f d o n o r - b o u n d excitons in selectively d o p e d G a A s / G a A 1 A s Q W s i n high m a g n e t i c fields. We believe that o n e o f the features o b s e r v e d in ref. [ 2 2 ] , n a m e l y , that the b i n d i n g energies o f the optically-active ( D +, X ) a n d ( D °, X ) states do n o t vary with m a g n e t i c fields u p to 27 T ( i n s t e a d o f i n c r e a s i n g with B) is a m a n i f e s t a t i o n o f the a b o v e m e n t i o n e d effect o f r e d u c i n g o f the e x c i t o n - i m p u r i t y int e r a c t i o n s for s - ( D ÷, X ) a n d s - ( D °, X ) in high m a g n e t i c fields. T h e s u p p o r t o f Professor S.G. T i k h o d e e v is gratefully acknowledged. I w o u l d also like to t h a n k Dr. S. H u a n t for d r a w i n g m y a t t e n t i o n to ref. [2] a n d for p r o v i d i n g m e with a copy o f ref. [ 2 2 ] .

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