wave functions of excitons in a homogeneous magnetic field B. The strength of B is arbitrary. ... analytical and may serve as a starting point for further applications. As for the ... angular momentum in the direction of B (see also. (31) in the ...
Condensed Matter
Z. Phys. B - CondensedMatter 66, 419M25 (1987)
Zeitschrift 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-mechanics 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 perturbation of the Hydrogen, the ordinary Rayleigh-Schr6dinger perturbation series does not converge. Sophisticated summation techniques had to be applied to overcome this difficulty. We refer to the publications of Avron, Herbst and Simon [1], Avron [2], Le Guillou 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 methods 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, Ruderman was the first one to argue that the ground-state binding-energy of Hydrogen in a B-field should behave as (ln B/Bo) 2 for sufficiently large B, where Bo is a suitable unit. Summarizing so far, we can state that the groundstate 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 Bassani [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. Gerlachet al.: Excitonsin a HomogeneousMagneticField
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 representatives, 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 applications. Instead, the data are only available in the computer memory, being used by the initiators of the calculation. Therefore, it is highly desirable to have reliable 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. Secondly, we produce interpolation-formulae, yielding upper bounds on the true energies. These bounds deviate only weakly from the lowest ones in [10-21]. A detailed comparison follows in part V.
cal mechanics, that P' is a conserved quantity). Obviously 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 Iel 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 - Ie 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)
II. S t a t e m e n t of 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.
The last term in H} can be removed by choosing KllB. In particular we assume B = (0, 0, B),
K = (0, 0, K).
Finally we introduce dimensionless variables by using (1)
ao:=4neo~h2/#e 2, h/ao, In (1), we denote the charge, mass, momentum and position of the electron by - ] e [ , ml, Pl, q~ ; analogously 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
(9)
(3)
(an indirect proof for [H', P']_ = 0 follows from (5) and (6) below; moreover, it is well known from classi-
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 according 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 "nonexciton" theory (see [I-5, 8-12, 14-20]). We come back to that point in part IV. Secondly, the continuum edge Ec of H is given by Ec/Ry=7.
(13)
Consequently, binding energies have to be calculated as
EB:=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 variational 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- Considered as a function of 7, the leading contribution to E(7) interpolates between the eigenvalues of a threeand a two-dimensional system. In our case these systems 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 extremely 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 contain variational parameters to make an interpolation possible. For the present problem the usefulness of (15) was earlier demonstrated by Pokatilov and Rusanov [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 zeropoint 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 quantum 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 description 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 systems 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/Ry
+1
= - - -
+7 2 ~/4~
7_ /~
1 02 ~t/ 8qo2
(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 asymptotics 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 e x p ( - 2 2 ~ q / 4 o . ) , where 2 2 / 0 - is at our disposal. This may need some explanations: an exact eigenfunction of H must reflect the interplay of Coulomb- and oscillator potential. For small 7 the character is more Hydrogen-, 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 unnecessary to represent the "interpolation-parameter" by a ratio of two, namely 22 and o. The following considerations will prove that this is useful. We anticipate that 2 is a scaling parameter. Putting all steps together, we substitute Om(~, q, ~0)=exp (im q~).(~ t]) Iml/2 9e 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~,, 2 2 (I m l +
4 { ~_ 0 r
1)/a
~-.~ ~ /{
ffo~ +~,~,t + \ l m l + l - 2 a
o~ 4 ~ +tl \
2 2a
(20)
,~.)c~ ~ 0ff
;}
(21)
~+~(2~--~)~/~)
+ (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: inspection of (22) shows that we could cancel the Coulombcontribution 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 parameter ~ 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 appropriately (m-dependent, as for details see the next part). In doing so, we find upper bounds to the corresponding exact eigenvalues.
(26)
Furthermore, 0 < n l , n2 < oo. Interestingly enough, the unperturbed energy (24) has already contributions of oscillator- and Hydrogen-type. This is characteristic 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 corresponding terms. They can all be expressed by contributions of type (24) or confluent hypergeometric functions ~(a, c; z), defined as
~(a, c;
Z):=~ 1
;dte-ztta-l(l+t)c
a
V(a) o
1.
E,,/Ry
