Z. Phys. B - Condensed Matter 66, 419M25 (1987) Condensed Zeitschrift Matter for Physik B

�9 Springer-Verlag 1987

Excitons in a Homogeneous Magnetic Field: A Modified Perturbation Approach

B. Gerlach and D. Richter

Institut fiir Physik, Universit/it Dortmund, Federal Republic of Germany

J. Pollmann

Institut ffir Theoretische Physik II, Universit/it Miinster, Federal Republic of Germany

Received December 17, 1986

We propose a specially adapted perturbational scheme to calculate the energies and wave functions of excitons in a homogeneous magnetic field B. The strength of B is arbitrary. In contrast to involved variational calculations, our final results are entirely analytical and may serve as a starting point for further applications. As for the energies, we find good agreement with previous work. Moreover we show that the well-known small-B and large-B asymptotics for the exact eigenvalues are both contained in our unifying formulas as limiting cases.

I. Introduction

The treatment of a Hydrogen-atom or (equivalently) an exciton in a homogeneous magnetic field B was one of the standard problems in early quantum-me- chanics and, surprisingly enough, remains to be a standard problem. In the last ten years the number of corresponding publications has sharply increased because of several reasons.

First of all, no analytical solution of the problem is available up to now. Moreover, interpreting the B-dependent terms in the Hamiltonian as perturba- tion of the Hydrogen, the ordinary Rayleigh-Schr6- dinger perturbation series does not converge. Sophis- ticated summation techniques had to be applied to overcome this difficulty. We refer to the publications of Avron, Herbst and Simon [1], Avron [2], Le Guil- lou and Zinn-Justin [3] and Silverman [4]. Due to their work, reliable results for the ground-state energy are known for magnetic field up to 10 l~ GauB ! Even more: Borel-summation techniques and related meth- ods have clarified the large-B behaviour of the ground-state for any given quantum number m of the angular momentum in the direction of B (see also (31) in the present text).

The study of the large-B behaviour has a history of its own, characterized by the key-word "adiabatic approximation". We mention the early work of Schiff and Snyder [5] and the more recent papers of Elliot and Loudon [6, 7], Ruderman [8] and Avron, Herbst and Simon [9]. To the best of our knowledge, Ruder- man was the first one to argue that the ground-state binding-energy of Hydrogen in a B-field should be- have as (ln B/Bo) 2 for sufficiently large B, where Bo is a suitable unit.

Summarizing so far, we can state that the ground- state energy for m--0 is well known. Moreover, the energies of ground-states for quantum numbers m :I= 0 can be discussed asymptotically. It is obvious that there exists a gap of knowledge: the energies and wave functions for higher states and intermediate fields B are not available.

Variational approaches have partially filled this gap and are in agreement with the work mentioned above. We refer to Larsen [10], Baldereschi and Bas- sani [11], Cabib, Fabri and Fiorio [12], Ekardt [13], Garstang [-14] (including an excellent review) and R6sner, Wunner, Herold and Ruder [15]. Particularly important for the present work were the papers of Pokatilov and Rusanov [16], Cohen and Kais [17],

420 B. Gerlach et al.: Excitons in a Homogeneous Magnetic Field

Gallas [18], Rech, Gallas and Gallas [19] as well as Chen, Gil and Mathieu [20]. We come back to that point in part III. Very recently, the possibility of chaotic behaviour in Hydrogen-B field systems has caused another boom in calculations. As representa- tives, we mention the publications of Wintgen and Friedrich [21] and Delande and Gay [22].

A practically unavoidable shortcoming of every involved variational calculation is the fact, that the resulting wave functions can hardly be communicated to other members of the physics community, which might be interested to have them for further applica- tions. Instead, the data are only available in the com- puter memory, being used by the initiators of the cal- culation. Therefore, it is highly desirable to have reli- able analytical expressions for the wave functions. It is the aim of this paper to show that these can be provided by a modified perturbation approach (part II, III). Lacking any rigorous results for the exact solution, the term "reliable wave function" has to be justified. We refer to the corresponding energies and stress two important properties of our results: firstly, the limiting cases of small and large magnetic fields are analytically reproduced (part IV). To the best of our knowledge, this cannot be found in any perturbational or variational scheme up to now. Sec- ondly, we produce interpolation-formulae, yielding upper bounds on the true energies. These bounds de- viate only weakly from the lowest ones in [10-21]. A detailed comparison follows in part V.

II. S t a t e m e n t o f the P r o b l e m

The Hamiltonian of an exciton, exposed to a constant magnetic field B, reads as follows:

H' := (Pa + [e[ A(q0)2/2 m~ + (P2-l el A(q2))Z/2m 2

--eZ/4rCeo e Iql--q21. (1)

In (1), we denote the charge, mass, momentum and position of the electron by - ] e [ , ml, Pl, q~ ; analo- gously we introduce [el, rag, P2, q2 for the hole. In addition, z is the dielectric constant of the background and A(q) the vector potential, taken to be

A (q) = -- q • B /2 . (2)

We do not include any spin-effect. Pauli-terms may simply be added.

It can be verified directly, that H' commutes with

e ' = Pl + P2 + l el (ql - q2) • 8 / 2 (3)

(an indirect proof for [H', P ' ]_ = 0 follows from (5) and (6) below; moreover, it is well known from classi-

cal mechanics, that P' is a conserved quantity). Ob- viously P' is the generator of a combined translation and a gauge transformation. To clarify its physical significance, we proceed in two steps: in a first one, we introduce center-of-mass and relative coordinates Q, P and q, p in the usual way. In a second step we perform a unitary transformation U, given by

g , = exp ( - i I el Q" (q x B)/2 h). (4)

This leads us to

H" := U- 1H' g=(m~P/M + p--lel q x B/2)2/2ml

+ (m2 P/M - p - l el q x B/2)2/2 m2

--e2/47Ceoeq (5)

P" := U -1P ' U = P . (6)

Interestingly enough, P' is transformed into the total momentum P. Clearly P commutes with H', proving that [H', P']_ = 0. To proceed towards a calculation of the spectrum of H", we may restrict H" to the subspace of fixed eigenvalue hK of P. We arrive at (see also Knox [23])

H~:=h2K2/2M+p2/2#+ lelZ(q x B)2/8 #

+ [el B'L/2#'-eZ/4neoeq

- I e I h K- (q x B)/M (7)

where L is the angular momentum and

M : = m 1-2i- m2, 1/# := l/m1 + 1/m2,

1/#' := 1/ml -- 1/mz. (8)

The last term in H} can be removed by choosing KllB. In particular we assume

B = (0, 0, B), K = (0, 0, K) . (9)

Finally we introduce dimensionless variables by using

ao:=4neo~h2/#e 2, h/ao, Ry:=eZ/8n~o~ao (10)

as units of length, momentum and energy. Subtracting the constant energy h 2 K2/2 M from H~, we get from (7)-(10):

H/Ry:=pZ-2/q+~,Z(q2~+q22)/4+TL3#/#'. (11)

The parameter 7 is the ratio of cyclotron and Rydberg energy,

7 :=l el B h/2# Ry. (12)

All further considerations are concerned with H ac- cording to (11). We add two remarks: firstly, #/#' may considerably deviate from 1. Insofar our Hamiltonian differs in a mathematically minor, but physically ira-

B. Gerlach et al. : Excitons in a Homogeneous Magnetic Field 421

portant point from the corresponding one in "non- exciton" theory (see [I-5, 8-12, 14-20]). We come back to that point in part IV. Secondly, the contin- uum edge Ec of H is given by

E c / R y = 7 . (13)

Consequently, binding energies have to be calculated as

E B : = 7 - E , (14)

where E is an eigenvalue of H. Summarizing, our problem can clearly be stated

as follows: solve the equaton H ~ = E ~ with H from (11).

III. The Modified Perturbation-Approach

In this part we introduce a perturbational treatment of the eigenvalue problem for Hamiltonian (11), which is valid for all values of 7. The approach incorporates some aspects of the rigorous solution as well as varia- tional ideas. As both facets are of decisive importance for the quality of the final results, we are now going to discuss them in some detail.

The unknown eigenvalues E depend on 7- Consid- ered as a function of 7, the leading contribution to E(7) interpolates between the eigenvalues of a three- and a two-dimensional system. In our case these sys- tems are the free exciton (three dimensional, 7--+0) and the two-dimensional oscillator (7 ~ oo). Despite this fact H is defined in three dimensions and the wave functions have to be normalized in N~ 3. It is well known, that under such circumstances an ex- tremely accurate treatment of the ground-state (m = 0) can be achieved by a trial wave function of the type

Off) = O3(r)" ~2( x, Y) (15)

where ~3 (r) models the three-dimensional, @2 (X, y) the two-dimensional limiting state. Both functions con- tain variational parameters to make an interpolation possible. For the present problem the usefulness of (15) was earlier demonstrated by Pokatilov and Ru- sanov [16], one of us (see Pollmann in [24]) and more recently in Refs. 17-20, mentioned above. We can profitably generalize the corresponding ideas: as H commutes with L 3, we can proceed similarly for the ground-state, belonging to any quantum number m of L 3 .

The property [H, L3]_ = 0 leads us to a second important point. If ~9,~(r) is an eigenfunction of H and L a with azimuthal quantum number m, the zero- point behaviour is governed by a prefactor (x 2

+72) Irnl/2 at least for 7 +0 . Let us now turn to 7 = 0 , where the system exhibits rotational symmetry in three dimensions. Using the standard description by spherical polar coordinates, the zero-point behaviour is now given by (x2+ y 2 + z 2 ) 1 / 2 , where I is the quan- tum number of angular momentum. One is tempted to argue that a smooth interpolation between the cases 7 = 0 and 7 + 0 is impossible. In fact this is not true; the problem is entirely an artifact of the descrip- tion in spherical polar coordinates. It is well known (see Landau [25] and in this context Ekardt [13]) that an alternative description of the Hydrogen case is possible by means of parabolic coordinates. The corresponding wave functions do have a zero-point behaviour of type (x2+ 72)1ml/2 (see also the following part, in particular (18)). Moreover, it is clear that parabolic coordinates are much better adapted to sys- tems with uniaxial symmetry (7 =~ 0).

Having this in mind, we substitute

x = ~ f ~ cos qo, y = ~ sin ~o, z = ( ~ - q ) / 2 (16)

into Eq. (11) and get

H / R y = - - - +1

+ 7 2 ~ / 4 ~ 7_ /~

1 0 2

~t/ 8qo 2

(17)

To determine eigenfunctions and eigenvalues of (17) due to the ideas outlined above, we proceed in several steps. In a first one, we separate the p-dependence by a factor exp (im ~0). In doing so, we restrict H to the subspace of a given eigenvalue m of L 3. In a second step we split off the zero-point asymptot- ics of the wave function by extracting a term (~l)lm[/2=(X2-~-y2) ItalIa. Thirdly, we compensate the oscillator potential 72~t//4 partially by a factor exp(-22~q/4o . ) , where 22 /0 - is at our disposal. This may need some explanations: an exact eigenfunction of H must reflect the interplay of Coulomb- and oscil- lator potential. For small 7 the character is more Hy- drogen-, for large 7 more oscillator-like. We model this behaviour by the above factor - exactly in the sense of (15). In particular, 7 ~ 0 will afford 22/o . ~ 0, 7 ~ oe needs 22/o.7 ~ 1. At the moment it seems un- necessary to represent the "interpolat ion-parameter" by a ratio of two, namely 22 and o. The following considerations will prove that this is useful. We antici- pate that 2 is a scaling parameter.

Putting all steps together, we substitute

Om(~, q, ~0)=exp (im q~).(~ t]) Iml/2

�9 e x p ( - 2 2 ~q/4o.). (p,,(~, t/) (18)

422 B. Gerlach et al.: Excitons in a Homogeneous Magnetic Field

into H ~b,~ = E m Ill m and arrive at

emq)m =(ho + h0 q~,, (19)

em,= Em/Ry - ? m I~/#' - 2 2 (I m l + 1)/a (20)

4 { 0 ~_ ~-.~ { ~ / ,~.)c~ h0== r ffo~ +~,~,t + \ l m l + l - 2 a ~ 0ff

o~ ;} (21)

4 2 ~ + ~ ( 2 ~ - - ~ ) ~ / ~ ) ~ +tl \ 2a

+ (72 - 2"/a 2) { q/4. (22)

We conceive (19)-(22) as a modified perturbation problem, h, being the perturbative correction of h o. To justify this, we stress the following points: inspec- tion of (22) shows that we could cancel the Coulomb- contribution 4(2-1) / ({+q) as well as the oscillator contribution (72-24/o-2) �9 ~t//4 by choosing 2 = 1 and 72= 24/a 2. Moreover, we introduce an additional pa- rameter ~ to achieve a partial compensation of the derivative-terms in h~.

Actually we do not fix 2, a and ~ that way. Instead we perform first-order perturbation theory, calculate the ground-state energy for an given value of m and minimize this energy by choosing the parameters ap- propriately (m-dependent, as for details see the next part). In doing so, we find upper bounds to the corre- sponding exact eigenvalues.

IV. R e s u l t s

)/ B . . . . 2 := l + ~ ( I m l + l ) ( n l + n 2 + l m [ + l ) . (26)

Furthermore, 0 < n l , n2 < oo. Interestingly enough, the unperturbed energy (24) has already contributions of oscillator- and Hydrogen-type. This is characteris- tic for wave functions as in (23), generalizing the one from (15).

Our next step concerns the perturbative correction fiE . . . . 2 of the energy, which is caused by h~. It is straightforward though lengthy to calculate the corre- sponding terms. They can all be expressed by contri- butions of type (24) or confluent hypergeometric func- tions ~(a, c; z), defined as

Z ) : = ~ 1 ; d t e - z t t a - l ( l + t ) c a 1. ~(a, c; V(a) o

(27)

In the remainder of this paper we concentrate on the ground-state for a given quantum-number m. In the above perturbation-approach, this corresponds to n~ =n2 =0 . Denoting the exact energy by E,,, we find

E, , /Ry <_ Bin, (28)

Bin:= [ m l + l /?go 22\ # 0272 2 ~ - + ~ - ) + ~ ym-~ 2Z(lm[ + 1) 2

2 a -- 2 2 -- a 2 3)2/2 2 4

(] m l + 1)2

7' (I m t + 1, 1 ; 2a/(I m l + 1) 2) 7'(Im [ + 1, 0; 2a/([ml + 1)z) " (29)

According to our previous considerations, the solu- tions of ho q~O = e o (po m are particularly important. They can be found by standard methods (see e.g. Landau- Lifschitz [25] and Erdelyi, Magnus, Oberhettinger and Tricomi [-26]). Using (18) and (20), we get

O~176 ~, ~o) = Aim . . . . .exp (im q~)-(22 ~ ,/)1,,I/2

�9 exp (-- 22 ~ r//4 o-). exp ( - 2 Am,l.2 (4 + q)/2)

�9 L~'~ I (2 B . . . . . 4)" LI.'~ I (2 B . . . . 2 t/) (23)

E ~ E,.,l~ ,2-- 22 { ( I m l + l ) / a + ( a / 2 a ) 2

-[(1 1))/(', +'. +'m' + 1)]" } + y m # / # ' (24)

where Nm . . . . is a normalization constant and

A . . . . . . ' = - - a / 2 a + B . . . . . (25)

It is interesting, that the energy bound B,~ depends only on 2 and a, but not on e. This is no longer true, if one considers excited states, which will be treated in a forthcoming paper.

The remaining task is to minimize B,, as a function of 2 and a. Analytical results can be found in the limiting cases 7 -+ 0 and 7 ~ oo. Making use of asymp- totic expansions for 7J(a, c; z), which can be taken e.g. from [26], we get the small- 7 expansion

B,, = - 1/(Iml + I) 2 + ymlz/la'

+ y2(Iml + 1)3(Iml + 2)/4 + O(73). (30)

For 7 -~ oo, one arrives at

Bm= (I m l + 1 + m #/#') 7 - (ln 7 - 2 In (In 7)-- q,,)2

- 8 1 n ( l n T ) - 4 ( l + q m ) + O ( ( l n [lnT])2/InT) (31)

where Iml 1

q m : = C + l n 2 + ~ v v = l

for m + 0 , q o : = C + l n 2 . (32)

B. G e r l a c h e t a l . : Excitons in a Homogeneous Magnetic Field 423

O ,4--

q C',l-

O N C S "

o e.i-

O

I -4.0

i0gl0 q//~00(X,J=0,Z) 2

,, ............. iiiiiiii

iiiiiii iiiiiii: I

-2.0

I ~1 I : : : , 5. i~ ~ {i i}}: ',i',i= i',i', i l ',', ' i',v i',i',.y, .',,', , , : , : , :U, lhh h',,

i , i J ! ,

.... / / " / ! ' ~ ~ i ,

i 0.0 2.0 4.0

Fig. 1 X

q

o N d

l O g 1 0 " @ O 1 0 0 ( X , y - - - ~ 0 , 8 2

q 2 1 I I

-10.0 -5.0 0.0 5.0 10.0

�9 "-,, ',i

o

~d

7 ',, ',. ~ iii

Fig. 2 X

C is the Euler-Mascheroni constant. We mentioned in the introduction, that (30) and (31) remain true for the exact energy E,~ itself (see Garstang [14] for 7--*0, Avron, Herbst and Simon [1] for ~--* oo). Our treatment provides the first explicit proof that both formulae are limiting cases of one unifying result.

Equation (31) shows a peculiarity of excitonic sys- tems. The positive quantity/~/#' may be considerably smaller than 1. Consequently, the leading term in Bm

I O g l 0 "@O400 ( X , y = 0 , Z ) 2 o.

N

- 6 0 . 0 - 3 0 . 0 0 .0 5 0 . 0 6 0 . 0

Fig. 3 X

Figs. 1--3. Contour plots for the decimal logarithm of the probability density. The values o f m a n d ~ are indicated. The small arrows denote the position of the maximum. Starting there and moving from one curve to the next, the logarithmic density decreases b y 1.

A 1.8454, 2 ~ z

-4- / / t

~ d -

/-. o 1-~.- ~ o . - ; - / 0

.0

X " 10. 0 z Fig. 4. Three-dimensional p l o t f o r the probability density, m = - 1,

7 = 0 . 1 . T h e b a c k - p l a n e s s h o w c u t s o f the plot with the planes x

: X m a x a n d Z = Z m a x

remains m-dependent also for m = - t m 1. No degener- acy appears as in the case # / # ' = 1.

We close this part with a selection of numerical results.

In the Tables 1 and 2 we present energy bounds Bm and the corresponding parameters 2, cr for m

4 2 4

Table 1. Energy bounds and variational parameters 2, a as functions of 7. m = 0 , # / # ' = 1. For comparison we list the data of [15 ]

y P resen t Res 15 2 - - 1 7 a w o r k

0.01 1.009950 1 .009950 5.04489 * 10 -09

0.02 1.019800 1.019800 8 . 0 1 5 4 4 * 10 - ~ 0.04 1.039201 1.039201 1.25902 * 10 - ~ 0.10 1.095048 1.095053 4 .52496 * 10 -05

0.20 1.180708 1.180763 5 .62700 * 10 - ~ 0.40 1.328766 1.329211 4 .64509 * 10 -03 1.00 1.659117 1.662338 3.23701 * 10 -02

2.00 2.035251 2 .044428 8.40428 * 10 - ~ 4.00 2 .541110 2 .561596 1.65941 * 10 -~

10.00 3.448433 3 .495594 3 .17716 * 10 T M

20.00 4 .351337 4 .430797 4.62443 * 10 -~ 40.00 5 .476534 5.602059 6 .31052 * 10 T M

100.00 7 .366627 7 .578100 8 .89136 * 10 T M

200.00 9 .151445 9 .453100 1.11075

400.00 11.288204 1 1 . 7 0 2 3 0 0 1.35542 1000.00 14.727013 15.324099 1.71535

2000.00 17.850120 18.608959 2 .01616

1000.28540

500.59765 251 .19332

102,90220

55.34456

33.66468 22 .98295

20 .74854

20.91779 23 .50760

26.95963 31,70837

40 ,22472

48 .66125 59 ,15630

76 .83520

93.65343

B. Gerlach et at.: Excitons in a Homogeneous Magnetic Field

Table 2. Energy bounds and variational parameters 2, cr as functions of Y. m= - - I , #/p'= 1. For comparison we list the data of [15]

7: Present Res 15 2 - 1 7 ~

w o r k

0.01 0 .269402 0 .269402 8.33963 * 10 . 0 6 168.44769

0.02 0 .287633 0 .287635 1.20792 * 10 04 86.79118

0.04 0 .320870 0 .320895 1 . 3 9 4 3 0 , 1 0 03 47 .94924

0.10 0 .401386 0 .401691 1.67054 * 10 ~~ 27 .91742 0.20 0 .499938 0 .501078 5 .76692 * 10 . 0 2 23 .29090

0.40 0 .639696 0 .642710 1.35356 * 10 -~ 22.67348 1.00 0 .905225 0 .913194 2 . 9 5 8 1 5 , 1 0 0x 25.40785 2 . 0 0 1.184714 1.199226 4 . 5 8 6 7 4 , 1 0 01 29 .60232

4.00 1.551020 1.575651 6.55785 * 10 T M 35.64654 10.00 2 .204689 2 .250845 9.68111 * 10 T M 46.90997

20.00 2 .859686 2 .930950 1.24376 58.43182

40.00 3 .685055 3 .792110 1.55416 73.14883

100.00 5 .093680 5.269480 2 .02010 98.72609 200.00 6 .446367 6 .694255 2 .41704 123.85615 400 .00 8.090201 8 .430228 2.85481 155.17922

1000.00 10.782086 11.276809 3 .50054 208.35397

2000.00 13.267779 13.903940 4 .04296 259.48568

0 I 0 ~ j

/

O L D :

o 7 ~ i _~t o J

0 1

"50

+ 1.0165 .0

10.0

Fig. 5; Three-dimensional plot for the probability density, m = -- 1,

7 = 1. Th e back-planes show cuts of the plot with the planes x = Xma x and z = Zma x

O

m = - 3 ,-2

m = - 2

m = - I

>..

LIJ

�9 0 III1114 I I l i l l i l i I I I l l l i l i [ I l i l l l l [ I I l l l l l l [ I I I l l g l [ I l i l l l l [ [

']- 10 -3 10 -2 10-' 10 0 10 ~ 10 2 10

7 Fig. 6. Present results (dashed lines) for the binding energy of Hydro- gen in a magnetic field, in comparison with the data of R6sner, Wunner, Herold and Ruder 1-15] (solid lines)

=0 , - 1 and different values of 7. We had to choose # /# ' - -1 for reasons of comparability. I f /~ /# '~ 1, the corresponding bounds can simply be found by adding ( / ( # ' - 1).Tin to Bin, given in the tables. The parame- ters 2 and o- behave as indicated in part III: for y ~-0 we find 2 ~ 1, 22/o--~0. One should notice that even 7- -2000 is far from being asymptotic in the sense of (31).

In Figs. 1-5 we show plots of the probability den- sity for variable y and m = 0 , - 1 , - 4 . As I~~ 2

is invariant against rotations in the x - y - p l a n e , we put y = 0. To guarantee comparability, the maximum of the probability density is normalized to 1 in all cases. Two main features are clearly visible: if (for m fixed) the value of 7 increases, t h e exciton states are more and more squeezed. This squeezing effect is mostly pronounced in the direction transverse to the field (in our case the x-direction). On the other hand, the "squeezing effectivity" of the magnetic field increases, if I m l becomes larger.

B. Gerlach et al.: Excitons in a Homogeneous Magnetic Field 425

V. Comparison with Previous Work

A short view into the quoted literature is sufficient to assure, that there exists a huge amount of numeri- cal data. We shall compare our results to those of R6sner, Wunner, Herold and Ruder [15], which in turn compare favourably with all previous work. We mention additionally, that extensive comparisons can be found in the paper of Le Guillou and Zinn-Justin [3].

Figure 6 shows our data in comparison with those from [15] for -3_<m_<0. Selected values are pre- sented in Tables 1 and 2.

We stated already in the introduction, that for many applications in solid-state physics, starting from excitons in a magnetic field, one needs analytical and tractable expressions for the wave functions. Our work is one attempt to proceed in that direction. The fact, that our results are in good agreement with far more involved numerical calculations, is both promis- ing and satisfactory.

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B. Gerlach D. Richter Institut fiir Physik der Universit~it Dortmund, Postfach 500500 D-4600 Dortmund 50 Federal Republic of Germany

J. PoUmann Institut f/Jr Theoretische Physik II, Universit/it Mtinster Wilhelm-Klemm Strasse 10 D-4400 M/inster Federal Republic of Germany