Ground and first excited states of excitons in a magnetic field

IL NUOVO CIMENTO VOL. 10 B, N. 1 11 I2uglio 1972

Ground and First Excited States of Excitons in a Magnetic

Field.

D. Cx~i~ (*) I s t i tu to di F i s i e a de l l 'Univers i t~ - P i s a

Gruppo .N'azionale S t ru t tura della Mater ia del C . N . R . - P i s a

:E. FABRI

Is t i t~ to di .Fisica dell' Univers i t~ - P i s a

I s t i tu to 2Vazionale di .Fisiea Nucleare - Sezione di P i s a

G. FIORIO

J~aboratorio M A S P E C del C.2V.R. - P a r m a

rieevuto il 10 Dieembre 1971)

S u m m a r y . - - An exact calculation of the ground state and of the first excited state with m ~ 0 and even par i ty of a hydrogenic system in a magnetic field is described, and results are given for the energy and the main features of the wave functions. For the excited state the shape of the nodal surface is given, and it is shown that in this case no con- tradiction exists between the noncrossing rule and the nodal-surface criterion for the connection of the levels in the low- and high-field regions.

1. - I n t r o d u c t i o n .

The p r o b l e m of a h y d r o g e n a t o m i n a m a g n e t i c field has b e e n the s u b j e c t

of m a n y t h e o r e t i c a l i n v e s t i g a t i o n s , on a c c o u n t of b o t h i t s i n t r i n s i c i n t e r e s t

for seve ra l p h y s i c a l p r o b l e m s , a n d the cons ide rab le m a t h e m a t i c a l dif f icul t ies

i t poses to a comple t e so lu t ion . A p p r o x i m a t e m e t h o d s h a v e b e e n deve loped

dea l ing w i t h t he l i m i t i n g cases of low or h igh fields, a n d va r i ous app roaches

(*) Present address: Depar tment of Physics, Michigan State University, East Lansing, Mich. 48823.

185

1 ~ 6 D. CABIB, E. FABRI and ~. FI0I~IO

have been sugges ted to br idge the gap b e t w e e n t h e m on the grounds of quali- t a t i ve a rgument s .

S c r u f f and S~YD]~ (~) s tudied the electronic s ta tes including the quadra t ic d iamagne t ic t e r m in f i rs t -order p e r t u r b a t i o n t heo ry for low values of the mug- net ie field.

IJx~]3 (~) considered the con t r ibu t ion to the ene rgy levels due to the t ransla- t ional mot ion of the a tom.

�9 "AFET~ ~J~EYES and A D ~ S (3) deve loped a va r i a t iona l calculat ion for the g round s ta te us ing a va r ia t iona l t r ia l func t ion wi th a Gauss ian shape, which makes i t good a t large values of the magne t i c field. A var ia t iona l approach was also adop ted b y P o ~ I L O V and t~vsA~ov (4) wi th an exponent ia l depend- ence of the t r ia l funct ions which allows t h e m to app roach the 1S, 2S and 2 P hydrogen wave func t ions for zero field.

I t can be shown (5.6) t h a t the m a t h e m a t i c a l fo rmula t ion of the p rob lem of the hydrogen a t o m in a magne t i c field appl ies qui te well, wi th in the accuracy of the expe r imen ta l resul ts , to o ther phys ica l s i tua t ions , such as to the isotropic exci ton in insulators and semiconductors , as well as to donor and acceptor impur i t i es ; the approx ima t ions involved are the ef fec t ive-mass t r e a t m e n t and the a s sumpt ion t h a t the re levan t bands are parabol ic and nondegenera te in the absence of spin. W ~ L I S and ]~0WLDEN (~) p e r f o r m e d a va r i a t iona l calcula- t ion on some exci ted s ta tes of donor impur i t i es in semiconductors . L~J~sE~ (~) gave a more accura te t r e a t m e n t of the g round s t a t e of donor levels in a magne t i c field and of the lowest - ly ing exc i ted s ta tes wi th m ~ 1 and m---- - -1 .

ELLIO~ ~nd ]~OIJDO~ (s) and FIClTSCKE (~) considered the p rob lem of exci tons in a magne t i c field and developed a pe r tu rba t ion -ad iuba t i c m e t h o d val id for high magne t i c fields. B&~DEnESC]~I and B&SSA~I (~o) improved these resul ts b y exac t ly solving the equa t ion of the ad iaba t ic m e t h o d a t h igh magne t i c fields.

(1) L. I. SC~IIFF and H. SNYDER: Phys..Rev., 55, 59 (1939). (2) W. E. LAMB jr.: Phys. Rev., 85, 259 (1952). (a) Y. 7~AFET, ]{. W. KEYES and E. N. ADAMS: Journ. Phys. Chem. Solids, 1, 137 (1956). (4) E. 1 ~ POKATILOV and M. M. RUSANOV: Sov. Phys. Solid State, 10, 2458 (1969). (5) R . F . WALLIS and H. J. BOWLD]~I~: Journ. Phys. Chem. Solids, 7, 78 (1958). (6) R . S . K~ox: Solid State Physics, edited by SEITZ and TI:]CNBULL, Suppl. 5 (New York, 1963). (~) D. lV[. LARS~N: Journ. Phys. Chem. Solids, 29, 271 (1968); H. 1~. FETTERM&I~, D. lV[. L~RS~, G. E. S~I~LMAN, P. E. TA~ENWALI) and J. WALDMAN : Phys. Rev. Lett., 26, 975 (1971); and private communication. (s) ]~. J'. ELLIOTT and R. LOUDON: Journ. Phys. Chem. Solids, 15, 196 (1960). (9) L. ]TRITSCHE: Phys. Star. Sol., 34, 195 (1969), reporting experimental results only in part reported by the authors in J. L. BREBNER, J. HALI'EnN and E. lV[OOSER: Helv. .Phys. Acta, 40, 385 (1967). (~o) A. BALD]~RESC~II and F. BASSAI~I: X International Con]erence on the Physics o] Semiconductors, Cambridge, Mass., August 17-21, 1970, published by the Atomic Energy Commission, p. 191; and private communication.

G R O U N D AND FIRST E X C I T E D STATES OF EXCITONS IN A MAGNETIC F I E L D 1 ~ 7

They also deve loped a pe r t u rba t i on -va r i a t i ona l m e t h o d val id for low fields a n d discussed how to join in the i n t e rmed ia t e region the resul ts ob ta ined wi th b o t h methods .

The p rob l em of joining ac tua l ly arises f r o m the different q u a n t u m number s b y which one can classify the energy levels in the low- and high-field l imits . ELLIOT and LOUDOSI (s) p roposed a connect ion p re se rv ing the n u m b e r of noda l surfaces, and F~ITSC~E (9) i n t e r p r e t e d the e x p e r i m e n t a l da ta b y BlUEBiRd, HALI'ElCN and lV[0OSER on the basis of this connect ion, whereus BOYLE and t tOW~,D (~1) discussed the da ta on hydrogen ic donors on the basis of the noncrossing rule s t a t e d b y var ious au thors (1~). Unfo r tuna t e ly , the two rules a re in comple te d i s ag reemen t except for the g round s ta te . ~ o r e recent ly , S R ~ D ~ et al. (13) p roposed a new fo rmula t ion of the nodal -surface rule, following which the 2S level goes in to the same high-field level foreseen b y the noncross ing rule. The p a p e r b y BAI~DE!~ESCtII and B~SSANI (~o) shows t h a t the four lowest high-field levels invo lved in the exei ton t rans i t ions are l ikely to connect wi th the four lowest zero-field levels according to the noncross ing rule. The agree- m e n t is good as fur as the b ind ing ene rgy is concerned. A fu r t he r inqu i ry appea r s however to be just if ied on account of the different me thods employed in the low- and high-field regions, and of the lack of knowledge of the exac t wave funct ion.

The purpose of the p re sen t work has been to ob ta in solut ions in the range f rom zero up to i n t e r m e d i a t e fields, i ndependen t of the choice of t r ia l funct ions . The resul ts for the ground s t a t e have a l ready b e e n publ i shed (*~). We r e p o r t he re the m e t h o d followed and the resul ts ob t a ined for the two lowest- lying levels: the energy eigenvulues are g iven for y r ang ing f rom 0 to 5 for the g round s ta te , f rom 0 to 2 for the exc i ted s t a te (2~ is the ra t io of the spacing of the L a n d a u levels to the effective l~ydberg of the exciton). As fa r as exe i ton t rans i t ions are concerned, only s ta tes whose wave funct ions do not van i sh a t the origin give rise to a firfite t r ans i t i on probabi l i ty . There fo re we h a v e res t r i c t ed our calculat ion to s ta tes wi th even p a r i t y and m ~ 0. The f irst exci ted s ta te in this set approaches the hydrogen ic 2S-s ta te as the magne t i c field goes to zero.

lr~ the ac tua l calculat ion we adop ted a sui table basis for the e igenfunct ions and reduced the t t a m i l t o u i a n to a f inite ma t r ix . The search for the e igenvalues is p e r f o r m e d th rough the m e t h o d of po lynomia l i t e ra t ion , deve loped b y two of us (15), which yields bo th an u p p e r und a lower bound. The exac t e igenvalue

(11) W. S. BOYLE and R. E. HOWARD: Journ. Phys. Chem. Solids, 19, 181 (1961). (le) Se6, ~or example, J. VON NEUMANN and E. P. WIGNER: Phys. Zeits., 30, 467 (1929); L. D. LANDAU and E. H. LIFSRITZ: Quantum Mechanics, Sect. 79 (Oxford, 1965). (13) M. SmNADA, 0. AKIMOTO, H. HASEGAWA and K. TANAKA: Journ. Phys. Soe. Japan, 2a, 975 (1970). (14) D. CABIB, E. FABRI and G. FIoRIo: Solid State Comm., 9, 1517 (1971). (15) E. FABRI and G. FIoRIo: Nuovo Cimento, 60 B, 210 (1969).

1 8 8 D. C A B I B , O. FIOI~IO and E. FABI~I

is therefore to be supposed to lie be tween the bounds we give in the Tables. A comparison of our results for the energy levels wi th those previously

published shows a v e r y fine ag reement wherever the resul ts of o ther calculations are supposed to give a good approximat ion .

Our procedure does not produce an analyt ic expression for the eigenfune- t ions; we are able, however, to compute numerical ly all quanti t ies of in teres t . We give in this pape r the values of the oscillator s t rengths (whose uncer ta in t ies are to be t aken as a sensible es t imate , and not as absolute bounds, which we cannot give in this case).

A discussion of the general fea tures of the e igenfunet ion of the exci ted s ta te (nodal surface, p robabi l i ty d is t r ibut ion, weights of the components of different angular momenta) shows tha t wi th increasing fields the wave funct ion con- t inuous ly changes in to the one of the second satelli te of the first Landau level, as also assumed by BALI)E~SC~I and ]3ASSA~I (10). In par t icular , the nodal surface gets more and more oblate, and in this sense the proposal of S~I~AD.( et al. (~8) appears to be correct .

2. - Mathematical formulation of the problem and n-merical methods.

The effective-mass approx imat ion leads to the following Schr6dinger equa- t ion for an electron and a hole in te rac t ing wi th each o ther in an insulator or semiconductor embedded in a un i form magnet ic field:

(1) ~fr - - V 2 - - + (x2+y~) r T

H ere we use the quant i t ies

* eh2 R * - #e4 B* 2/~eR* a o ~ - - ~__,

# e ~ ' 2 e 2 h 2 ' he

US the uni ts of length, energy and magnet ic field; y is the magnet ic field in these uni ts . The z-axis has been t aken in the d i rec t ion of the field. ~b is the envelope funct ion, depending only on the re la t ive electron-hole co-ordinate r.

I t can be shown (le) tha t the allowed e lect romagnet ic dipole t rans i t ions are those leading to a final s ta te whose envelope funct ion does no t vanish a t the origin; the t rans i t ion probabi l i ty is propor t iona l to [~b(0)]~.

The Hami l ton ian ~%f defined in (1) is inva r i an t unde r ro ta t ions about the z-axis and under space inversion, therefore i ts eigenveetors have definite p a r i t y and definite z -component of angular m o m e n t u m m. We are in te res ted only

(le) R. J. ELLIOTT: Phys. l~ev., 108, 1384 (1957).

G R O U N D AND F I R S T E X C I T E D STATES OF E X C I T O N S I N h M A G N E T I C F I E L D 189

in the solutions wi th m = 0 and even pa r i t y because only these funct ions

do not vanish at the origin. Then we can write down the solution ~( r ) as a suitable combina t ion of spherical harmonics wi th even 1 and m-~ 0:

(2) r 0, ~) = ~ r Y~o(0, ~) �9 e v e n

By subs t i tu t ing (2) into (1) we obta in an equivalent l inear sys tem of differ- ential equat ions of the form

(3) ~ ]~n,q~,(r) : EqS~(r) ,

where

(4) ~.,= -- ~r+ r"

The mat r ix elements wu, couple the eqs. (3) and express the in terac t ion with

the magnet ic field

(5) w~, = -~r 2 dY2 Y,o(O, ~c) sin s 0 Y,,o(0, ~0).

Clearly, wu, = wt,z and we have

(6)

W [ z ( / + l ) ] w~t -~ ~-r ~ 1 - - ( 2 / - - 1 ) ( 2 / + 3)

~,~ ( / + 1)(z + 2) W l fl-{- 2 -~ - - - " ~ r 2 _ _ _ ,

(2/-}- 3)%/(2/-4- 1 ) ( 2 / + 5)

wz~, = 0 in all o ther cases ,

as can be seen by remember ing tha t sin s 0 is a l inear combinat ion of I7oo and IQo.

For ease of numer ica l computa t ion it is be t t e r to use a radial co-ordinate defined in the in te rva l I ---- [0, 1] and re la ted to r b y

(7) r = a ~ / ( 1 - - ~ ) .

The scale factor a is chosen in order to give a good representa t ion of the region

where the wave func t ion is impor tan t . This choice however is far f rom critical. ~ o w we define

(s) u~(~) = ~ ( r )

190 D. CABIB, :Eo FABI~I and G. FIOI~IO

for eve ry l; eqs. (3) becom e

(9) ~ -~ a ~ d~ ~ + r ~ On,-+- w w uz , -~ E u t

The scalar p r o d u c t on the ~-co-ord ina te is to be defined as

(lO) 1

(u, v) = a" f u * ( # ) v ( # ) ( 1 - - ~)-'d~:. 0

The b e h a v i o u r of the wave func t ion a t small r is d ic ta ted b y the cent r i fugal po ten t ia l , hence uz goes as ~. Using eq. (8) we then ob ta in the re la t ion

(11) 1 ~duo~ ~(o) = ~ \~-1~oo"

To p e r f o r m the numer i ca l calculat ion we a p p r o x i m a t e the se t of different ia l equa t ions (9) b y a finite se t of difference equat ions as follows:

i) d iv id ing the i n t e rva l I on the ~-axis in to a finite n u m b e r N of equal segments ,

ii) a p p r o x i m a t i n g the second de r iva t ive b y a cen t ra l f ive-point formula,

iii) cu t t ing off the expans ion (2) and the t t a m i l t o n i a n ~%f' a t a max- i m u m Z.

Then u~(~) is now an ( N - - 1 ) - c o m p o n e n t vec to r ( the poin ts ~-----0 and ~ ~ 1 are no t t a k e n into account ; t hey cor respond to r---- 0 and r = c~ and the re we know uz(~)= 0). The scalar p r o d u c t of two vec to rs is defined b y (10), w i th the in tegra l rep laced b y a sum over the divis ion poin ts of I and d~ replaced b y ~ = l / N :

2I--1

(12) (u, v) = (~a, ~. ( 1 - ~,)-~u*(#~)v(r i = 1

with ~1 = ~, ~ = 2 0 , ... , ~2/-1= ( N - - l ) 0 . The H a m i l t o n i a n ~%f' is also reduced to a finite m a t r i x /~/u'~' of order

( N - - 1 ) ( L / 2 + 1 ) whose H e r m i t i c i t y requires the use of cent ra l fo rmulae to a p p r o x i m a t e the second der iva t ive . The cent ra l n-poin t fo rmula (n odd) intro- duces (17) an error of order (~-1, therefore one should a t t e m p t to use a fo rmula

(17) Handbook of Mathematical Funct ions, edited by ~I. ABI~AMOWITZ and I. A. STEGUN, National Bureau of Standards, Chap. 25.

G R O U N D AND F I R S T E X C I T E D STATES OF E X C I T O N S I N A M A G N E T I C F I E L D 191

with big n in order to get b e t t e r precis ion wi th lower X. B u t a problem arises when one applies a difference formula wi th q, > 3 at the boundar ies of I . F o r example, the five- (seven-, ...) po in t formula applied to the po in t ~1 = (~ requi res the value(s) of the func t ion at the point(s) ~_~=--(~ (~_~=

- -28 , ...) which should be ob ta ined by means of a k ind of ext rapola t ion, bu t avoiding to in t roduce errors of order lower than (5 ~-~. B y using the sys- t em (9) in the l imit r-->0, one obtains the analyt ic cont inuat ion for t h e / - c o m - ponent uz of the e igenfunct ions:

(13a) Uo(-- (~) = - - Uo(8)[1 + 2a8(1 + aS)] + 0 ( 8 ' ) ,

(lab) uz(-- (~) ~- - - u~(8) + 0((5 4) for l # 0 .

This ex t rapola t ion employs only the value of the uz in the first poin t of defi- n i t ion uz(8), so we can take i t in to account by only modifying the e lements Hlzaz if we use the f ive-point formula. The correct use of a seven- or more-point formula requires more complicated modifications of the t t ami l ton ian and seems no t to give appreciable advantages�9

Squar ing the expression (11) we obta in the oscillator s t rength. So we need to calculate the first der iva t ive at the origin wi th respect to ~ of the componen t Uo(~), which is known only in discrete points. We can use a forward difference formula

(14) u" = ~ ~ - -~ +-~ Uo + 0(8,)

Alternat ively , we can express u"(O), u ' (0 ) , u~V(0) as funct ions of u'(0) using the sys tem (9) in the l imit r - + 0 ; by subs t i tu t ing into a Taylor expansion for u0(5) abou t the origin, we obta in

(15) Uo(8) = Uo(8). &l. [ ~3 a ]-1

�9 1 - - a ~ + ~ (2a--6--Ea) + -i8 (2a~E--a2--6aE + 1 2 a - - 1 8 ) + o(~,).

Actually, we computed the oscillator s t reng th wi th bo th methods , ge t t i ng ve ry similar results.

In order to approx imate even b e t t e r the eigenvalues of the t rue problem we repea ted the calculat ion for several values of X(8, 16, 32, 64) and extrapo- la ted these results as follows (is). We took three different values of 5V, each obta ined b y doubling the previous one, so tha t if E l , / ~ , E8 are the corresponding

(is) E. FABRI and G. FIORIO: Nucl. Phys., 141h, 325 (1970).

192 D. CABIB, E. FABRI and G. FIORI0

values of the calcula ted energy, we have

(16a) ~1 = E ~- 2SE4 -~ 210e5 ,

(16b) J~2 = E + 24e4 + 25E5 ,

(16c) E3 = E + e4 + e5 ,

where E is a q u a n t i t y which differs f rom the t rue energy b y t e rms of order ~e and ~4, ~5 are the effects of four th- and f i f th-order errors on Es. F r o m eqs. (16) we ge t the ex t r apo la t ed va lue of the ene rgy b y e l iminat ing e4 and e5:

(17) = ( 5 1 2 E ~ - - 4 8 z ~ + ~ , ) / 4 6 5 .

A useful re la t ion connec t ing El , ~2, E~ unde r the a s sumpt ion t h a t 4e5 << e4 is

(18) E 3 - - E2 ~.~ __1 J~2-- E1 - - 16 "

The two ex t rapo la t ions wi th 8, 16, 32 po in t s and 16, 32, 64 po in t s can be com- p a r e d and fo rmula (18) can be used to t e s t whe the r errors up to th i rd order have been e l imina ted and the sixth- and h igher -order errors are u n i m p o r t a n t ; for example i t t u rns out t h a t a b a d choice of the cons tan t a in (7) would lead to a d i sag reemen t be tween the two ex t rapo la t ions because i t affects the rep- r e sen t a t i on of the funct ion in d iscre te points . I f we solve s y s t e m (16) for s4 and s5 we ge t an e s t ima te of the goodness of the ex t rapola t ion .

Because of t i m e l imi ta t ions the p rocedure ske tched above was a b b r e v i a t e d for the exc i ted s ta te : the calculat ion was only p e r f o r m e d for s = 8, 16, 32, and there fore the compar i son be t ween the two ex t rapo la t ions was suppressed.

3. - Calculation o f the two lowes t -energy states.

The p rob l em is now reduced to the search for the e igenvalues and the cor- r e spond ing e igenvectors of a finite matrix//~z.~.z. (i, i '=- 1, 2, . . . , . N - - l ; l, 5'---- = O, 2, .. . , L).

F i r s t of all we no te t ha t the m a t r i x / / s u r e l y has an u p p e r bound A to i t s ene rgy s p e c t r u m because i t is a f ini te-order ma t r i x . We c o m p u t e d A as descr ibed elsewhere (18) b y the i t e ra t ion p rocedure ] . = ~ / ' ] for eve ry pa i r iS, _N. The func t ion ] is a n y ini t ia l t r ia l funct ion , on which the final resu l t does not

depend. The i t e r a t ion m e t h o d requires t h a t the t t a m i l t o n i a n m u s t h a v e only one neg-

a t ive eigenvalue. I n comput ing the g round s ta te , the above condi t ion is fulfilled

G R O U N D A N D F I R S T E X C I T E D S T A T E S O F E X C I T O N S I N A M A G N E T I C F I E L D 19S

simply by subt rac t ing f rom the t t ami l ton ian a lower bound to the nex t eigenvalue and eventua l ly by adding i t to the calculated eigenvalue. Once the ground s ta te has been calculated, the same procedure is followed wi th the nex t eigenvalue bu t the t r ia l funct ion is or thogonMized to the eigenfunct ion of the ground state , thus s imulat ing the condi t ion requi red b y the i te ra t ion method. The lower bounds to the eigenvalues have been approx imate ly es t imated f rom previous works.

I f ] is the ini t ia l t r ia l func t ion and In is t ha t a t the n- th step of the i tera- t ion, the following relat ion holds (15) when I]]nll > [Jill:

(#- n)IIfII" <19) iij, - i]iii

where # and #~ are the expec ta t ion values of H on ] and f~. We used this formula to stop the i t e ra t ion at a fixed precision. For various reasons this precision is not complete ly a rb i t ra ry : first of all the round-off errors of the computer in t roduce a l imi ta t ion on the n u m b e r of digits one can obta in for the eigen-vMue, bu t this is of no impor tance in our case; the re levant l imi ta t ion is the one con- cerning the exci ted state, which obviously cannot be computed wi th the same precision as the ground star% since any error of the la t te r re f lec t s onto the former because of the orthogonalizat ion. So if one wants a given precision for the exci ted s tate , he must reach a b e t t e r precision for the ground state. Inci- dentally, we note t ha t the or thogonal izat ion mus t be periodical ly r epea ted during the i t e ra t ion in order to cancel f rom ]~ the growing component along the ground state.

The cut-off in t he / - expans ion (2) is re la ted to the precision requi red for the eigenvalues in the following way. The i t e ra t ion is r epea ted b y successively adding one angular component , thus obtaining a decreasing sequence of energy values, unt i l the difference be tween two successively calculated values of /~ is much less t ha n the requi red precision. I t turns out t ha t for this purpose the exci ted s ta te m a y require a grea ter number of components than the ground s ta te ; in this case the orthogonMizat ion is pe r fo rmed to a ground-s ta te func- t ion wi th the missing components set equal to zero.

The procedure of repea t ing the i t e ra t ion wi th increasing Z does not waste t ime because the e igenfunct ion obta ined in the previous i te ra t ion can be employed as tr ial funct ion simply by adding a null componen t ; this is in general a good t r ia l funct ion and the convergence is v e r y fast .

Doubling the number s is the nex t step, and the ent i re procedure is repea ted in order to per form the ex t rapola t ion (17). This t ime the last com- p o n e n t of the t r ia l funct ion (and the ent i re t r ial funct ion for the first Z) is obta ined b y quadrat ic in te rpola t ion of the same component(s) for the previous N.

Le t us now summarize the approximat ions involved in our calculation

13 - I1 Nuovo Cimento B.

194 D. CABIB, E. FABRI a n d G. FIORIO

in order to give a clear under s t and ing of how an e s t ima te of the errors has been made , a t leas t as far as the ene rgy eigenvMues are concerned.

F i r s t , the re is the f ini teness of the n u m b e r of radia l po in ts /V. The precision is enhanced th rough the ex t rapo la t ion descr ibed above ; a t the same t i m e the ma in cont r ibu t ions (0(~ ~) and 0(~5)) to the error can be evaluated .

Second, the re is the reduc t ion of the inf ini te series in l to a f ini te sum. Since the error cannot readi ly be guessed in this case, we adop ted the pol icy of going on to higher l 's unt i l the con t r ibu t ion of the las t componen t became negligible.

Third, we h a v e the in t r ins ic er ror of the i t e ra t ion procedure . F o r i t we have bo th u p p e r and lower bounds , as we h a v e shown. The to ta l e r ror a t t r i - b u t e d to our resul ts t akes in to account all t h ree contr ibut ions .

A much more difficult t a sk has been to assigu an er ror to the oscil lator s t rength . ~ o defini te ex t rapo la t ion is possible, nor can useful bounds be g iven along the i t e ra t ion . Therefore we r e so r t ed to compu t ing ]~(0)l~ f r o m b o t h eqs. (14) and (]5), and infer r ing an e s t ima ted er ror f rom the compar i son of the results , and f rom the behav iou r of t h e m wi th increasing 2V and iS.

4. - N u m e r i c a l results and d i scuss ion .

The ground s t a t e was calcula ted for severa l values of y be tween 0 and 5, as shown in Table I ; the n u m b e r of angu la r componen t s necessary to ge t a f ixed precision on the energy e igenvalue is an increasing funct ion of y, as we could have expec ted b y considering t h a t the magne t i c field in the z-direct ion squeezes the func t ion a round the z-axis. F o r 7----5 we used seven angu la r componen t s cor responding to a m a x i m u m L = 12. We s topped the i t e r a t i on so as to get a precision of a b o u t 10 -4 re la t ive to the in te rva l be tween the energies of the ground and first exci ted s ta te . The weights of the var ious angular componen t s are given in Table I I for some cases. Table I shows t h a t our resul ts are lower t han those yie lded b y all the var ia t iona l methods , and t h a t t h e y agree wi th those of B~DE~ESCnI and BASSA•I (lo) a t low and high fields, b a t are g rea t ly i m p r o v e d in the i n t e rmed ia t e region.

I n Table I I I we give the oscil lator s t r eng th of the g round s ta te in un i t s of the zero-field value (1/~). Our resul ts are in much b e t t e r ag reemen t wi th Pt~ (4) and L (7) t h a n wi th the ad iaba t i c m e t h o d (lo).

The first exc i ted s ta te requi res a much g rea te r n u m b e r of angular componen t s , especia l ly for high values of y. There fore we compu ted this s t a t e only for 7 up to 2, wi th a m a x i m u m of 13 angular componen t s (~ = 24). As a consequence of the increasing t ime reques t of the compu ta t i on wi th higher ~, the precis ion ob ta ined was k e p t decreasing, bu t the absolute e r ror neve r exceeds 2 .10 -3. The weights of the angula r componen t s are g iven in

Table V.

GROUND AND FIRST EXCITED STATES OF EXCITONS IN A MAGNETIC FIELD 195

TABLE I. -- Energy o/ the ground state in units o] the e]]ective l~ydberg R * = #e4/2s2h 2. (The resul ts of Y K A (5) and of P R (4) have been r e c o m p u t e d , for t he sake of compar i son , s t a r t ing f rom t h e ana ly t i c solut ion.)

y Y K A (a) P R (4) L (~) B B v (10) B B a (i0) P r e s e n t work

0.1 - -0 .84443 - - 0 . 9 9 5 0 5 - -0 .995 - - 0 . 9 9 5 0 8 • 5-10 -5

0.2 - -0 .83956 - -0 .98071 - -0 .981 - - 0 . 9 8 0 7 6 4 - 5-10 -5

0.3 - -0 .81097 - - 0 . 9 5 8 1 7 - - 0 . 9 5 8 3 5 - - 0 . 9 5 8 - - 0 . 9 5 8 4 1 • 6-10 -5

0.4 - -0 .78364 - -0 .92876 - -0 .927 - - 0 . 9 2 9 2 3 4 - 5-10 -5

0.5 - -0 .75054 - - 0 . 8 9 3 6 4 - - 0 . 8 9 - - 0 . 8 9 0 - - 0.89447 :j:: 7 .10 -5

0.6 - -0 .71252 - - 0 . 8 5 3 7 4 - -0 .84 9 - - 0 . 8 5 4 9 4 4 - 7 .10 -5

0.7 - -0 .67032 - -0 .80981 - -0 .81 - -0 .80 5 - - 0 . 8 1 1 4 2 4 - 7-10 -5

0.8 - -0 .62449 - -0 .76243 - - 0 . 7 5 8 - - 0 . 7 6 4 5 7 4 - 7.10 - s

0.9 - - 0 . 5 7 5 5 4 - -0 .71207 - -0 .70 8 - - 0 . 7 1 4 7 3 4 - 7 .10 -5

1.0 - - 0 . 5 2 3 8 6 - - 0 . 6 5 9 1 2 - -0 .66217 - - 0 . 6 5 5 - -0 .7 1 - - 0 . 6 6 2 4 1 4 - 9 .10 -5

1.5 - -0 .23416 - - 0 . 3 6 4 5 0 - - 0 . 3 7 - - 0 . 3 7 - - 0 . 3 7 0 7 6 4 - 7 .10 -5

2.0 0,09190 - - 0 . 0 3 5 2 5 - - 0 . 0 4 - - 0 . 0 4 - - 0 . 0 4 4 5 0 4 - 1 1 . 1 0 -5

2.5 0.44192 0.31707 0.31 0.31 0.304904- 6 .10 -5

3.0 0.80911 0.68596 0.68 0.67 0.670874-12"10 -5

4.0 1,5798 1.4589 1.45 1.4384 4-10.10 -5

5.0 2.3842 2.2648 2.2433 2.24 2.2392 4-10.10 -~

TABLE II . - Weights o] angular components in the ground state, /or several values o] y, obtained at the last iteration step with the maximum 5V. No a t t e m p t has been m a d e to e s t i m a t e t he errors nor to e x t r a p o l a t e t h e weights , h o w ev e r t he i r f luc tua t ions dur ing t h e i t e ra t ion sugges t t h a t two d ig i t s are s ignif icant excep t for t h e few las t c o m p o n e n t s .

1

0.1 0.2 0.5 1.0 2.0 5.0

0 1.0 0.99990 0.99848 0.993 28 0.979 84 0.944 54

2 4 .10 -5 0.00010 0.00151 0.006 59 0.019 28 0.050 54

4 7.10 -5 2 .10 -7 0.000 O1 0.00014 0.00082 0.00436

6 1.10 -~ 3 .10 -e 0.00005 0.00048

8 3 .10 -e 0.00007

10 0.00001

12 1.10 -6

1 ~ D. CABIB, E. FABRI a n d G. FIORIO

TABLE I I I . - Oscillator strength o] the ground state in units o] ~-1a~-3. ( T h e r e s u l t s o f Y K A (8) a n d o f P R (4) h a v e b e e n r e c o m p u t e d , f o r t h e s a k e o f c o m p a r i s o n , s t a r t i n g f r o m t h e a n a l y t i c s o l u t i o n . )

7 Y K A (a) P R (a) L (7) B B a (lo) P r e s e n t c a l c u l a t i o n

0.1 0 .244 1 .010 1.009 4- 1 - 1 0 -8

0 .2 0 .254 1.037 1.034 4- 1 . 1 0 -a

0 .3 0 .271 1.078 1.067 1.071 4- 1 - 1 0 -8

0 .4 0 .291 1.229 1.115 4- 1 - 1 0 -8

0 .5 0 .315 1.187 1 . 1 6 4 1 1 . 1 0 -8

0 .6 0 .341 1.247 1 .219 4- 2 . 1 0 -a

0.7 0 .369 1.312 1.275 4- 2 10 -8

0 .8 0 .398 1.379 1 .334 :[: 2 10 -8

0 .9 0 .429 1.448 1.390 4- 10 10 -8

1.0 0 .460 1.519 1 .435 4 .6 1.449 4- 2 10 -8

1.5 0 .628 1.883 4 .85 1.759 4- 2 10 -a

2 .0 0 .808 2 .264 5.2 2 .076 i 1 10 -8

2 .5 0 .996 2 .650 5.6 2.401 :L 1 10 -8

3 .0 1.191 3.041 5 .95 2.731 4- 1 10 -8

4 .0 1 .599 3 .836 6.7 3 .39 4- 10 10 -8

5 .0 2 .026 4 .645 4 .15 7.5 4 .07 4- 10 10 -3

TABLE I V . - Energy o/ the ]irst excited state in units o] the e]]ective l~ydberg.

B B v (lo) B B a (lo) P r e s e n t w o r k

0.1 - - 0 . 1 9 5 0 - - 0 . 1 9 6 2 • 1 . 1 0 -4

0 .2 - - 0 .0954 - - 0 .0979 • 1 - 1 0 -4

0 .3 0 .0067 0 .0033 • 1- l 0 -4

0 .4 0 .1040 0 .127 0 .1017 • 1 - 1 0 -4

0 .5 0 .2006 0 .217 0 .1984 • 1 - 1 0 -4

0 .6 0 .2967 0 .310 0 .2945 4- 1 . 1 0 -4

0 .7 0 .3927 0 .404 0 .3905 i 2 . 1 0 -4

0 .8 0 .4889 0 .4985 0 .4865 4- 2 . 1 0 -4

0 .9 0 .5854 0 .592 0 .5828 4- 2 . 1 0 -4

1.0 0 .6821 0 .687 0 .6793 -4- 3" 10 -4

1.5 1 .169 1 .1636 4- 5" 10 -4

2 .0 1.657 1.651 4- 2 . 1 0 -a


TABLE V. - Weights o] angular components in the ]~rst e~cited state, /or several val~e8 o/ ~. See also t h e c a p t i o n of Tab l e I I .

l y

0.1 0.2 0.5 1.0 2.0

0 0.971 32 0 .82695 0 .42950 0 .25451 0.165 13

2 0.028 25 0.162 56 0.452 25 0.477 09 0.406 77

4 0.000 43 0.009 75 0.095 01 0.183 41 0.239 05

6 2 . 1 0 -6 0 .00068 0 .01851 0 .05858 0 .10950

8 0.000 06 0.003 73 0 .01813 0.046 66

10 4 -10 -6 0.000 78 0.005 64 0.019 45

12 0.000 17 0.001 78 0 .00804

14 0.000 04 0.0O0 57 0.003 28

16 5 . 1 0 -6 0.000 20 0 .00133

18 0.00007 0.00051

20 0 .00002 0 .00019

22 3 -10 -6 0 .00006

24 0.00001

TABLE VI . -- Oscillator strength o/ the ]irst excited state in units in ~-la~-S.

y B B a (16) P r e s e n t c a l c u l a t i on

0.1 0.160 • 1 .10 -3

0.2 0.179 i 1-10 -a

0.3 0.165 l 1 .10 -a

0.4 0.146 • 1 .10 -a

0.5 0.133 ~ 1 .10 -a

0.6 0.124 • 1 .10 -a

0.7 0.118 • 1-10 -s

0.8 0 . 1 1 5 • 1 .10 -8

0.9 0.113-V 1-10 -a

1.0 0.23 0.113 ~ 2 . 1 0 -a

1.5 0.25 0.114 ~= 2 . 1 0 -8

2.0 0.26 0.120 • 5 -10 -a

1 9 8 D . C A B I B , E . F A B R I and G. F~OR~O

Our energies are aga in lower t h a n the ad iaba t ic and var ia t iona l ones b y BB (lo). The oscillator s t r eng th of this level is g iven in Table V I ; i t a lways s tays near the zero-field value, 0.125 in our uni ts , and exhibi ts a m a x i m u m a t y __ 0.2 and a m i n i m u m at y ~ 1 .

I n order to ge t ~ b e t t e r insight into the first exc i ted s t a te we have ske tched in Fig. 1 the p robab i l i t y dens i ty and the shape of the noda l surface for several values of ~. One can easi ly see f rom the F igu re t h a t the p robab i l i t y dis t r ibu- t ion ac tua l ly shr inks a round the z-axis, b u t does no t ca r ry the noda l surface

,Z §

T 2 ~Z

". .... <:~

~ = 0 =0.2 =0.5 =2.0

Fig. 1. - Cross-section of the wave function of the first excited state in a plane through the z-axis. The curve marked . . . . . is the trace of the nodal surface. The density of the hatching is proportional to the absolute value of ~.

wi th i tself ; on the con t ra ry , the l a t t e r f la t tens in the z-direct ion and bulges on the equa tor ia l p lane. This suggests t h a t the l imi t ing shape of the nodal surface a t h igh fields should be a pa i r of p lanes or thogonM to the z-axis as sugges ted b y SKI~AD• et al. (13), r a t he r t h a n a cyl inder , as a s sumed b y ]~LLIOTT and LOU~ON (8). There fo re the first exc i ted s t a t e should go, wi th increas ing field, f r o m a 2• hydrogen ie s t a te in to the second satel l i te of the first L a n d a u level, (0, 0, 1 +) in the no ta t ion of SKI~ADA et al. (13), according to the noncrossing rule and to the calculat ions of BB (lo).

The second absorp t ion p e a k in GaSe (~.13) should there fore be ass igned to the first L a n d a u level (N = 0) r a t he r t h a n to the second ( N = 1) as sugges ted b y F~ITSCI~E (g). Unfor tuna te ly , this resul t is of no help in i n t e rp r e t i ng the o ther peaks ar is ing in the cont inuous s p e c t r u m of the electron-hole sys tem. The expe r imen ta l da ta avai lable a t p r e sen t (7.13) cover the low-field region ( ~ < 0.25), where the ad iaba t ic a p p r o x i m a t i o n is no t just if ied; on the o ther hand, the absorp t ion peaks exhib i t a Landau- l ike p a t t e r n which canno t be expla ined wi th a p e r t u r b a t i o n approach . A sa t i s fac to ry model , account ing


for the observed transition probabilities in this region has not yet been pro-

posed.

T h e a u t h o r s w i s h t o t h a n k P r o f . G. F . BASSANI f o r h i s i n t e r e s t i n t h i s w o r k ,

f o r h i s s u g g e s t i o n s a n d f o r r e a d i n g t h e m a n u s c r i p t .

�9 R I A S S U N T O

Si dese r ive u n ealeolo e sa t to dello s t a t o f o n d a m e n t a l e e del p r i m o s t a t o ecc i t a to con con m = 0 e p a r i t s -~ di u n s i s t em a i d r o g e n o i d e in e a m p o m a g n e t i c o , e si r i p o r t a n o i r i s u l t a t i r e l a t i v i a l l ' ene rg ia ed alle ca ra t t e r i s~ iche p r inc ipa l i delle funz ion i d ' o n d a . D a l l a f o r m a del la superf ic ie noda le del p r i m o s t a t o ecc i t a to si fa v e d e r e che, in ques to caso, n o n e'~ c o n t r a d d i z i o n e t r a la regola d i n o n i n t e r s ez ione ed il c r i te r io di c o n s e r v a z i o n e delle superf ic i noda l i hel lo s t ab i l i r e la connes s ione f ra i l ivel l i a piecol i e g r a n d i va lo r i del c a m p o magne t i eo .

OCHOBHOe H llepB0e B o s 6 y ~ e l m o e COCTOIIHI4L~I 3KCHTOHOB B Mal['HHTHOM none .

Pe3ioMe (*). - - OrmcbmaeTca TOqHOe BB1HHCneHHe OCHOBHOFO COCTO~IHH H nepBoro BOS= 6y~-~eH~lOFO COCTO~HHH C ~ - - 0 H NOnO;q~HTeYIBHO~ ~teTROCTBIO ~YLq Bo~opo~HO~ CHCTeMbI B MarHHTHOM zone . ]-[pHBO~ITC~I pe3yn/~TaWbI )UI~I 3 a e p r a a n OCHOBrlBIe oco6ermocwa BOnaOBbIX qbyHKtm~. ~ n a BO36ymaennoro COCTOHHr~ npaBO~TCZ qbopMa y3noBo~ Ho- BepxHocTI/I. IIoKa3BmaeTc~, HTO B STOM cYucqae He cymecTByeT npOTHBOpeh~HH Meanly npaBH~OM nenepeceqerm~ H KpnTepHeM y3noBo~ nOBepXHOCTH ~Y~ CB$I3H ypOBHeH B 06naCTH HH3KHX H BBICOK~X Hone~.

(*) 1-Iepeeec)eno pec)ataluert.

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Ground and first excited states of excitons in a magnetic field