Maximal symetrization and reduction of fields: application to wavefunctions in solid state nanostructures

Maximal symetrization and reduction of fields:
application to wavefunctions in solid state nanostructures

S. Dalessi [    M.-A. Dupertuis [ Laboratory of Physics of Nanostructures, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
19 août 2019

A novel general formalism for the maximal symetrization and reduction of fields (MSRF) is proposed and applied to wavefunctions in solid state nanostructures. Its primary target is to provide an essential tool for the study and analysis of the electronic and optical properties of semiconductor quantum heterostructures with relatively high point-group symmetry, and studied with the formalism. Nevertheless the approach is valid in a much larger framework than theory, it is applicable to arbitrary systems of coupled partial differential equations (e.g. strain equations or Maxwell equations). This general MSRF formalism makes extensive use of group theory at all levels of analysis. For spinless problems (scalar equations), one can use a systematic Spatial Domain Reduction (SDR) technique which allows, for every irreducible representation, to reduce the set of equations on a minimal domain with automatic incorporation of the boundary conditions at the border, which are shown to be non-trivial in general. For a vectorial or spinorial set of functions, the SDR technique must be completed by the use of an optimal basis in vectorial or spinorial space (in a crystal we call it the optimal Bloch function basis (OBB)). The full MSR formalism thus consists of three steps : 1) explicitly separate spatial (or Fourier space) and vectorial (spinorial) part of the operators and eigenstates, 2) choose, according to the symmetry and well defined prescriptions (e.g. specific transformation properties), optimal fully symmetrized basis for both spatial and vector (or spin) space, and 3) finally apply the SDR to every individual scalar ultimate component function. We show that with such a formalism the coupling between different vectorial (spinorial) components by symmetry operations becomes minimized and every ultimately reduced envelope function (UREF) acquires a well-defined specific symmetry. The advantages are numerous: sharper insights on the symmetry properties of every eigenstate, minimal coupling schemes (analytically and computationally exploitable at the component function level), minimal computing domains. The formalism can be applied also as a postprocessing operation, offering all subsequent analytical and computationnal advantages of symmetrization. The specific case of a quantum wire (QWRs) with point group symmetry is used as a concrete illustration of the application of MSRF.

73.21.-b, 78.67.-n
preprint: APS/123-QED

Present address: ]Computational Biology Group, Department of Medical Genetics, University of Lausanne, Rue du Bugnon 27, CH-1005 Lausanne, Switzerland ]

I introduction

The study of low dimensional solid state nanostructures is a very interesting and promising domain. Indeed a good knowledge of electronic and optical properties of nanostructures like metallic Girard (2005) or semiconductor Yoffe (2002) nanostructures, or photonic crystals Sakoda (2005), is essential for many applications in advanced lasers, photonics and telecommunications. New truly quantum applications like quantum cryptography also make extensive use of semiconductor quantum heterostructures like quantum wells, quantum wires and quantum dots Stevenson et al. (2006); Young et al. (2009). The quality of semiconductor quantum wires and dots have been extensively improved during the last fifteen years, and one is now able to produce high quality structures with higher and higher point-group symmetries (e.g. quantum dots Kako et al. (2004); Andreev and O’Reilly (2000)). In such a case a group-theoretical approach is usually the most powerful tool for describing the effects resulting from symmetry on the electronic states, their optical properties in particular. However the problem is rather complicated since in heterostructures one must take into account both the underlying microscopical crystalline structure and the mesoscopic heterostructure confinement potential.

The theoretical study of low symmetry effects in semiconductor heterostructures (like quantum wires with symmetry, e.g. T- and V-shaped quantum wires Akiyama (1998); Runge (2002)) is already well developed Dupertuis et al. (2000) and has lead to fundamental conclusions regarding their electronic and optical properties. First electronic and excitonic states can be labelled with respect to their characteristic transformation properties under symmetry operations, second rigorous and important selection rules were readily obtained on the basis of such a classification, useful even in a low symmetry case Dupertuis et al. (2000); Dupertuis (2000). The effects of lateral confinement to the polarization anisotropy were largely studied Vouilloz et al. (1998); Sercel and Vahala (1991, 1990); Bockelmann and Bastard (1991); Citrin and Chang (1991). However, it should be pointed out that up to now only very little work has been devoted to higher symmetries (called here HSH: High Symmetry Heterostructure), for example structures Hartmann et al. (1999); Michelini et al. (2004), or even  Tronc et al. (2004). A particularity of HSH is to allow the existence of symmetry-induced degenerate eigenstates, related to irreducible representations (irreps) with dimension greater than one Cornwell (1984); Altmann and Herzig (1994), and which display a much more complex behavior under symmetry operations. Symmetry-induced degenerate eigenstates play an important role in the generation of entangled photon pairs from quantum dots Young et al. (2009).

The electronic structure of semiconductor heterostructures are very often studied in the frame of the envelope function approach for heterostructures Bastard (1988), with at least four bands when describing the valence band. In such a frame the different envelope functions (components of the spinorial eigenstates) become entangled under symmetry operations: their shapes are therefore mutually coupled, and their behavior is complex. Up to now there has been very few attempts to try to use an Optimal Bloch function Basis (OBB). In Dupertuis et al. (2000) we tried to rely on the concept of an ’Optimal Quantization Axis (OQA) direction’ (A Bloch function basis which diagonalizes the component of angular momentum in the chosen optimal direction). In fact we shall show in the following that such a method is of limited interest: it is optimally adapted only in very low symmetry cases like structures! For Quantum Wires (QWR) with a higher symmetry group, the previously defined OQA direction may only be an improved choice, not the optimal choice.

In this paper, we propose the optimal and systematic solution to this problem: a general Maximal Symmetrization and Reduction (MSRF) formalism, perfectly adapted to the study of scalar or spinorial HSH problems. We will show how to find true OBBs which minimize the coupling between envelope functions, and which truly maximize their individual symmetry. Moreover we will show how to systematically compute the whole solution on a reduced minimal domain. With the improvement of growth techniques increasingly high symmetries can indeed be produced enhancing the need for novel tools allowing to fully take into account the symmetry properties and to significantly simplify, theoretically and numerically, the understanding and the computation of electronic and optical properties. We originally developed the MSRF formalism to study a quantum wire (QWR), and therefore we shall often use it as an example, but one should stress that the method is general, applicable to other groups and to widely different cases since in fact its possible scope of application is much wider: it could be applied in its full generality to arbitrary tensorial fields obeying a set partial differential equations characterized by any given point group symmetry. In particular the method is independent of the number of coupled bands kept in the problem, and independent of the specific terms kept, provided the global symmetry is conserved. For example we could easily take into account interface terms Foreman (1993); Lassen et al. (2004) or strain terms in an eight-band approach Stier et al. (1999); Boucaud and Sauvage (2003), but in case of strain we would also need to treat in the same way the elasticity equations for the strain tensor (this could be done most conveniently by post-symmetrization of the elasticity calculation, see end of Section VI). The MSRF method is also independent of dimension, it applies equally well for two or three spatial dimensions, i.e. to QWR or to quantum dot (QD) heterostructures with a given point-group symmetry.

Let us now shortly present the heart of the MSRF formalism, which is threefold. First for every quantum states one performs an explicit separation of the spatial character (3D orbital motion, eventually treated in Fourier space) and of the field character (e.g. spinorial). Second one selects optimal fully symmetrized bases, both for spinorial space (the Optimal Bloch function Basis (OBB)) and orbital space, which minimize the coupling between different spinorial components. These two ingredients allow to obtain every spinorial component of the field as a sum of symmetrized scalar functions of spatial coordinates. Third for every irrep one identifies minimum sets of independent parameters (orbital reduced domains) which form the systematic Spatial (or Fourier) Domain Reduction technique (SDR) and which allow to obtain reduced Hamiltonians with respect to reduced domains, and systematically minimally reduce the computing requirements.

The advantages of the new MSRF formalism are manifold. Indeed besides the possibility of performing SDR we shall show that there are also many advantages from the analytical point of view: first the Hamiltonian operator usually takes a simpler form in the adapted fully symmetrized basis (OBB), second the spinorial components of eigenstates (as well as the components of any operator in the spinorial basis) can be treated in a similar way and easily decomposed into fundamental parts to which single group irreps can be associated (and for which ”sub-selection rules” can be applied at an intermediate calculational level). In this way particularly simple analytical expressions can be obtained for certain operator matrix elements, which allow to find, for example, new analytical ratios in the polarization anisotropy that were previously unnoticed in the numerics Zhu et al. (2006). Further insight can also be gained from the fact that this symmetry-based technique simplifies the expression of coupling matrix elements. Most notably weak symmetry breaking mechanisms can be understood more deeply at the analytical level Dalessi et al. (2009). From the numerical point of view, the systematic SDR technique will enable one to solve independently for every irrep on a minimally reduced solution domain. The SDR technique not only allows to find the boundary conditions at the boundary of such a domain, shown below to be non-trivial, but it also allows to eliminate the need to explicitely care for them! It should be pointed out that the MSRF method can also be used as a post symmetrization technique on numerical results obtained without taking into account any symmetry. In such a case it not only allows to classify all eigenfunctions and symmetrize them within the OBB, and benefit of an in-depth symmetry analysis, but in all subsequent computations symmetrized wavefunctions on reduced domain can then be used, which may still represent a further significant potential gain.

Let us now detail the necessary procedures of the proposed MSRF technique.

For scalar problems, like the single band spinless conduction band Hamiltonian, it reduces to our systematic SDR procedure. The SDR procedure involves two fundamental steps: first the spatial domain must be decomposed into a minimal number of disjoint sub-domains which map into each other through symmetry operations (including borders as separate domains), second the set of domains and wavefunctions must be projected on the relevant irreps. This last step allows to find the critical geometrical features of all states by identifying for every irrep the minimal independent parts of any function of a given symmetry, to which a corresponding reduced subdomain can therefore be associated. At this stage non-trivial boundary conditions can be derived if necessary. The same procedure can then be carried out for all relevant functional operators like the Hamiltonian. The reduced Hamiltonian reflects directly the coupling between different sub-domains and does incorporate automatically the restrictions implied by non-trivial geometrical boundary conditions.
For spin dependent problems, like the Luttinger Hamiltonian describing the valence band in diamond semiconductors, or the much used eight-band Hamiltonian Boucaud and Sauvage (2003), the OBB basis functions must be first found. They transform like an irrep of the (double) group, and allow to block-diagonalize the corresponding matrix representation of the double-group. By choosing the OBB, one enforces a minimal coupling between different spinorial function components under symmetry operations. Since every component can then be decomposed in a simple way into scalar envelope functions labelled with single groups irreps, the SDR technique is applicable to every component, and reduced Hamiltonians can be found for every double-group irrep.

In section II we shall recall in more details essential results obtained in low symmetry heterostructures which are needed to understand the HSH challenge and establishing the basis for the development of the MSRF formalism. In Section III, transformation laws in both ordinary space and spin space are studied, together with fundamental group-theoretical results needed for the definition of the fully symmetrized OBB basis and for the separation of spinorial and spatial parts. These goals are attained in Section IV). In the next two Sections, we first develop the SDR formalism for single group spinless (scalar) functions (Sec. V), and in the second we apply the SDR to the symmetrized envelope functions created by the OBB (Sec. VI). In addition we show how the technique leads to reduced Hamiltonians, and how it can be used as a post-symmetrization technique. Finally, in Section VII we demonstrate how selection rules can be applied at an intermediate level - a specific feature of the MSRF formalism - to compute the matrix elements of operators. As a result we find novel strong analytical results (polarization anisotropy in structures) which can be interpreted with the help of the generalized Wigner-Eckart theorem for point groups. Finally, in Section VIII, an outlook is given on other symmetry groups (the hexagonal group, an approximate zone-center symmetry group , the (commutative) subgroup of the rotation group and the group). The potential of the MSRF formalism for different problems is shortly described in Section IX as well as its relationship with the most close works found in the litterature on heterostructures which exploit symmetry.

Ii Envelope-function theory of low symmetry heterostructures and the HSH challenge

The MSRF formalism developed in this paper is built upon specific techniques previously developed for low symmetry heterostructures. The main goal of this section is not only to introduce the basic envelope function Hamiltonians for the conduction and valence band that will be used throughout the paper, but also to recall fundamental results on low symmetry heterostructures Dupertuis et al. (2000) which are at the origin of MSRF. We will also show explicitly the limitations of these techniques which do not allow to reach maximal symmetrization for higher symmetry heterostructures, which thus represent a main challenge. These results will form an essential basis for the systematic study of transformation laws developed in Section III and the development of the cornerstone of the MSRF in Section IV.

ii.1 Introduction to envelope function models

Multiband Hamiltonians are extensively used for the study of the electronic structure in semiconductor heterostructures Bastard (1988). Such models allow to introduce all the relevant physics close to a high symmetry point of the band-structure, whilst keeping maximum simplicity. For many applications in III-V zincblende semiconductors like it is possible to treat separately the conduction and the valence band problems. Nevertheless the method presented in the following is generalizable to more complex multiband schemes which treat simultaneously the coupling between these bands, like eight-band or fourteen-band Hamiltonians.

Let us also assume a heterostructure translation with translation invariance in some spatial directions and which are confining in the remaining directions, like quantum wells or quantum wires (see Fig. 1), and split the coordinate system into and respectively.

For the isolated conduction band the approach gives rise to the simple effective-mass model when one ignores spin-splitting terms. In the case of QWRs or QWs the conduction band Hamiltonian operator is defined by its action on any electron state . In the r representation, this amounts to apply the following differential operator


on the electron wavefunction , which is a simple scalar function of position. Here the perpendicular gradient is of course related to the confined directions, and translation invariance of implies a translation invariant confining potential and effective mass , both independent of , and the appearance of the corresponding good quantum number , which can be interpreted as the electron momentum in the free directions. Note that we have used for convenience in Eq. (1) the somewhat clumsy notation , which comprises differential operators like and variables like and , keeping in mind that, in the following, eventual transformations on the argument of must also be applied consistently to . Let us also denote , the eigenstates of associated with the eigenvalue .

For the valence-band let us use the minimal four band Luttinger Hamiltonian Luttinger and Kohn (1955); Luttinger (1956); Bastard (1988); Fishman (1995) which is required when one wants to have a good estimate of the optical polarization anisotropy. Such model also provides a fairly good description of the QWR valence subband energy dispersion close to the zone-center.

In its original form the bulk Luttinger Hamiltonian can be written as Luttinger and Kohn (1955); Luttinger (1956)


where denotes all cyclic permutations of the indices () of the matrix representations of the components of quasi-angular momentum . The corresponding envelope function Hamiltonian for the QWR valence band reads:


where is the diagonal part incorporating the effect of the heterostructure confinement potential, here is the wave vector along the wire, and is the kinetic part as given by the bulk Luttinger Hamiltonian where . As a result the kinetic part can always be put in the following standard form Fishman (1995):


In this equation the matrix elements are -dependent partial differential operators obtained from the Luttinger quadratic polynomials (2). Eq. (4) may also involve spatially-dependent Luttinger parameters corresponding to the spatial composition dependance of the heterostructure. The eigenstates of can be considered as 4D spinorial fields (related to the Bloch function basis at the top of the valence-band), and each component of this field is a -dependent scalar field of , the so-called envelope functions.

At this stage a few fundamental comments are in order. First the actual form of the coefficients is a function of the Bloch function basis chosen. The standard form Fishman (1995), corresponding to the main crystal directions , is obtained using standard matrix representations Messiah (1969) of . However for heterostructures whose main symmetry elements differ, one should usually define a rotated cartesian frame with a new -axis oriented along adapted directions, e.g. see Fishman (1995) for directions, such that the corresponding implicit Bloch function basis diagonalizes the new component of the quasi-angular momentum.

It is however important to point out that the shape of each envelope function is basis-dependent, even though the expectation value of any physical quantity remains basis-independent! This point, which will be illustrated in the next subsection, was not explicitely recognized for a long time, and complicates a lot the question of the symmetry of the envelope functions in the presence of valence band mixing.

ii.2 The effects of symmetry on the envelope functions

It is well known Bassani and Parravicini (1975) that the existence of a symmetry group for an Hamiltonian allows to classify each kind of eigenstates, to deduce their degeneracies, and to specify their possible transformation properties, however in the case of a spinorial set of envelope functions, the analysis of the symmetry of every individual envelope and its transformation properties has not been addressed and requires much more work (Section III). Let us give here an intuitive justification for this extensive effort. We shall first discuss the effects of a single symmetry plane, already quite well-known (see Dupertuis et al. (2000), and subsequent works Dupertuis (2000); Dupertuis et al. (2002); Marti et al. (2005a, b)). Then we shall enlighten in detail the difficulties involved in applying these results, based on rather elementary concepts, to higher symmetries. The discussion will both develop intuition and a new understanding of the difficulties involved, demonstrating the need for MSRF.

ii.2.1 A low symmetry case: symmetry

Let us now assume a heterostructure with the simplest symmetry, like the typical V-shaped QWR shown in Fig. 1, with a single symmetry plane. The corresponding symmetry point group is . Let us take the coordinate system such that . The symmetry of the heterostructure with respect to the plane implies that and are both invariant with respect to the symmetry plane operation .

Figure 1: View in perspective of a typical V-shaped QWR, with coordinate system. The central part is typically pure , whilst the surrounding bulk part is typically with .

The wavefunction profiles of the first two eigenstates of the stationary Schrödinger equation are shown on Fig. 2.

(a) (b)
Figure 2: Contour plots of conduction band envelope functions of the first two electronic states in a typical V-shaped QWR
(a) Even function (ground state); (b) Odd function (first excited state).
The V-shaped QWR potential well is shown with a doted line, the vertical quantum well with is also visible.

It is easy to show that the symmetry of the structure implies that the eigenstate symmetry can be labelled as being either “even” or “odd”. Indeed we see on Fig. 2 that the ground state wavefunction is strictly even with respect to the symmetry plane , whilst the first excited state wavefunction is strictly odd. The higher states will all display one character or the other unless there would be an accidental degeneracy, which would then allow an accidental mixing within the degenerate subspace. To summarize: the eigenfunctions all obey one of the following transformation rules under , the operation of reversing a wavefunction with respect to the -symmetry plane


It is also obvious that relations (II.2.1) do translate into stringent conditions for the properties of the wavefunctions on the symmetry axis:


Such relations are very useful because they can be used as boundary conditions to reduce the domain of solution on the left or right halfplane, which is therefore the natural reduced domain of solution of the stationary Schrödinger equation in this case. In most solution schemes, e.g. real space methods like finite element (FE) or finite differences (FD) approaches, it is easy to obtain odd and even solutions separately by solving two times the eigenproblem with different Dirichlet/Neumann boundary conditions on the symmetry axis boundary. Let us shortly have a group-theoretical approach: the even and odd wavefunctions shown in Fig. 2 correspond respectively to the and irreducible representations (irreps) of the group. and are simply new group-theoretical labels meaning “even” or “odd”, and no further insight arises. Nevertheless our considerations related to the symmetry of conduction band wavefunctions in symmetry - despite their trivial aspect - will prove important to better appreciate the differences occurring for valence-band envelope functions.

The case of the valence band eigenstates in symmetry is much more complex because of their (four-dimensional) spinorial character, this is why point group theory will immediately become an unvaluable asset. It tells us Altmann and Herzig (1994) that the spinorial eigenstates bare two labels again, but this time corresponding to the double group irreps instead of and . The spinors bearing these labels display a much more subtle and disturbing behaviour: there is not any more any simple intrinsic symmetry for each of the envelope function components [where ]. Their individual symmetry may indeed depend on the basis chosen, and, although there are two kinds of eigenstates, there is no such simple geometrical interpretation of the symmetry of such states by words of every-day life like “even” or “odd”. The role of their more complex label is precisely to convey the nature of these more complex transformation laws under mirror symmetry.

Let us now illustrate this behaviour with the valence band eigenstates of the V-shaped QWR shown in Fig. 1. With the standard Bloch function basis used in the early works on the subject Bockelmann and Bastard (1991); Vouilloz et al. (1998), the ground state envelope functions are those of Fig. 3, where one clearly sees that none of the envelope functions is either perfectly symmetric or antisymmetric with respect to the symmetry plane! This “standard” choice of Bloch function basis was at the time guided by the fact that the shape of a V-shaped QWR is close to a deformed quantum well, and it was indeed reasonable since many qualitative features of the optical absorption spectrum could be understood Bockelmann and Bastard (1991); Vouilloz et al. (1998) on the basis of “quantum well light and heavy holes”. This is why this basis is the one diagonalizing the component of pseudo-angular momentum aligned with the crystalline direction (note: with the choice of labels as in Fig. 1 this actually correspond the vertical direction , i.e. ).

Figure 3: Contour plots of valence band envelope functions of the ground state spinor (symmetry ) in a typical V-shaped QWR, when the Bloch function basis diagonalizes corresponding to the crystal direction (same QWR as in Fig. 2). (a-d) function components respectively ().

Even if the envelope functions of Fig. 3 are not symmetric, intuitively one still expects some symmetries induced by the QWR symmetry. Indeed a closer analytical look reveals that there are still some symmetry relations linked with or eigenstate. They can be formulated as follows:


Clearly such symmetry relations, which only couple envelope functions, cannot enforce the individual symmetry of every envelope functions in the spinor, and are nevertheless awkward from the numerical point of view since they do not allow to reduce the domain of solution on the half-plane as in the spinless case!

The clue to this problem was found in Dupertuis et al. (2000) by choosing a different Bloch function basis which diagonalizes the component oriented along the crystalline direction defined in Fig. 1. In such a case one could find novel envelope functions associated with every quantum state, with the following symmetry for or states respectively:


To illustrate this we display in Fig. 4 the contour plots for the same ground state as in Fig. 3. Although seemingly different, this new envelope function representation for the eigenstate carries exactly the same physics, i.e. gives the same expectation values for all physical observables.

Figure 4: Contour plots of valence band envelope functions of the ground state of symmetry in a typical V-shaped QWR, when the Bloch function basis diagonalizes the component corresponding to the crystal direction (same QWR as in Fig. 2). (a-d) function components respectively ().

It should be mentioned that the basic reason for the -behaviour of the envelope functions in Eqs. (8) compared to Eqs. (7), which might seem surprising at first, can in fact be explained in a very intuitive way by looking at the behaviour of angular momentum components through a planar reflection : indeed the sign of the in-plane components of the angular momentum are reversed, while the perpendicular component is conserved, i.e.


Therefore it is obvious that if one uses a Bloch basis diagonalizing the component of the pseudo-angular momentum perpendicular to the symmetry plane (i.e. ), every envelope function will be mapped onto itself (either in a symmetric way or antisymmetric way), whilst the and components will be mapped onto each other if one diagonalizes . To show that the envelope function spinors linked with the two double group irreps and have opposite alternating parity in Eqs. (8) requires a more detailed analysis. However we prefer to relegate this discussion after the presentation of the general theory, since most properties will become obvious.

ii.2.2 From low to high symmetries: , and higher

We have thus shown that a careful choice of basis allows, in the case of , to symmetrize individual envelope functions. Would this be possible in the case of higher symmetry? We intend now to clearly demonstrate that the approach suggested in Ref. Dupertuis et al. (2000), where we introduced the concept of Optimal Quantization Axis (OQA), has limits, motivating the more elaborate MSRF approach. We shall now take quantum dots as simple examples.

Let us start with , the next higher symmetry depicted on Fig. 5. A practical example is pyramidal QDs which are have the shape of a rhombus-based pyramid. We shall denote the two perpendicular symmetry planes and for compatibility with the axes used in this paper. Since is a sub-group of , we could use the same basis diagonalizing , and reduce the problem to the half-domain. However it would be highly desirable to use the additional symmetry with respect to to further split the domain of solution.

Figure 5: Perspective view of a QD

No problems for electrons, but problems would arise for holes, since a 4 band model would be required, and from our previous discussion of symmetry we immediately see that if we would take the spin quantization axis along , the spinors envelope functions would become alternatingly even/odd with respect to , but with respect to would necessarily obey a symmetry relation coupling and (c.f. Eq. (7). Oppositely if one would have chosen the basis, the symmetry relations with respect to would have become badly behaved. Therefore it is apparently never possible to obtain symmetric envelope functions in the two directions simultaneously, and solve on the half domain in the two directions!

There is a rather simple explanation at a more fundamental level for this different behaviour. For the electron symmetry, described by the single group, and commute, and therefore one can in principle diagonalize the two operations simultaneously, and get simultaneous good quantum numbers linked with them. For the hole spinorial symmetry, described by the double group, and do not commute, indeed the general commutator can be written as:


showing that when is half-integer (here ), it is never possible to diagonalize simultaneously both symmetry operations (such a fact can also be related to the appearance of a 2D irrep for the double group and the properties of its corresponding unitary matrix representation ). As a result of this analysis we suggested in Dupertuis et al. (2000) that the optimal basis was naturally the one diagonalizing the projection of angular momentum along the third perpendicular axis, allowing to treat and on an equal footing, and diagonalizing the rotation . However a closer inspection reveals that no solution on the quarter of the domain is yet allowed in by this idea, this will only become possible with MSRF.

Even more challenging is the next higher symmetry: symmetry, with three symmetry planes like in Fig. 6. Such a symmetry is also of practical interest Hartmann et al. (1999); Michelini et al. (2004).

Figure 6: Perspective view of a QD

The reference axes of the crystal, and our labels, are shown in more detail in Fig. 7, together with the three vertical symmetry planes .

Figure 7: Axes and cross section of the QD

In addition to the three improper rotations (vertical symmetry planes), called , the group includes as additional symmetry operations two rotation of () and the identity ().

The group displays two 1D irreps and , and one 2D irrep (), even for the single group, leading to two basis function (partner functions) for the subspace related to degenerate eigenvalues. There is a supplementary difficulty linked with this 2D irrep, in particular the corresponding 2D matrix representation explicitly depend on the basis functions. The simpler 1D single group irreps and are respectively even and odd with respect to all the symmetry planes. Therefore, for electrons it is straightforward to see that one can compute easily and eigenstates via the solutions on 1/6 of the domain by imposing respectively Neumann or Dirichlet boundary conditions on the two symmetry planes. For the degenerate irrep one can choose the two basis function such that they are either even or odd with respect to one of the symmetry plane (mirror), for instance , but it will not be possible to diagonalize simultaneously any two of the mirrors at the same time, for instance and , since they do not commute, as can be seen in the multiplication table (see annexe A). Therefore it is not obvious how to use the mirror symmetries to solve on a reduced domain smaller than one half. One may conclude that in a heterostructure the problem that appeared only for valence-band holes in the case of symmetry already appears for spinless electrons: it is not possible to solve with this technique on the most reduced domain, which, in , should be smaller than one half of the full domain! For holes in there is also a 2D faithful self-conjugated double group irrep (), but also two 1D mutually conjugated irreps (Altmann and Herzig (1994).

Finally all these questions become more severe in higher symmetry, like the hexagonal symmetry (c.f. perspective view of a QD in Fig. 8) which is also of practical interest Simeonov et al. (2006); Tronc et al. (2004). A characteristic of this group is that the double group displays only 2D degenerate irreps. Although such structures have been discussed to some extent in the literature with the help of group theory Tronc et al. (2004), the symmetry properties of the envelope functions have never been studied and discussed. With MSR it is possible to approach systematically all higher symmetries, and for the results will be given in subsection VIII.1.

Figure 8: Perspective view of a QD

To summarize, we can identify a limit between low and high symmetry groups in our sense: the appearance of a 2D irrep, which is a manifestation of the non-abelian character of groups like the dihedral groups () that we have considered. Such irreps with dimension greater than 1 complicate a lot the question of the symmetry of the basis functions, making it non-trivial.

Finally, the concept of spin quantization axis direction, linked with optimal pure rotations of the original spinorial basis, is not a suitable concept to tackle such higher symmetries (with spin, already !). Clearly maximal symmetrization of the envelope functions could not be achieved, and computation on a reduced domain was not enabled. In the following, a radically new approach for HSH is presented which will fulfill these goals. The optimal spinorial basis is obtained by a more general unitary transformation corresponding to the reduction (block-diagonalization) of the spinorial representation and related to the choice of double group labelled basis functions.

In the next chapter we shall start developing the MSRF formalism from beginning, by looking at transformation laws in “orbital” and “spin” space. Whenever needed the general theory will be illustrated by the typical case of a QWR (2D problem), either for the spinless conduction band, or for the valence band, again with the four-band Luttinger Hamiltonian. Numerical examples were worked out in real space with a FE approach incorporating linear elements only.

Iii General transformation laws, in ordinary and spinorial space

The study of transformation laws under symmetry operations is a prerequisite for the efficient use of symmetry and group theory explicitely on a given problem. One needs to know exactly how conduction and valence band envelope functions do transform under symmetry operations. This is of course related to the corresponding Hamiltonians presented in Sec. II. In this section we introduce in details transformation laws, which are also a prerequisite for the development of our new theory, independently of any symmetry consideration. For clarity we treat seperately scalar functions and spinors. In the last part of the section only we introduce symmetry, and look at the resulting constraints on envelope functions encountered in theory.

iii.1 Transformation laws

Let us first introduce transformation laws for simple spinless scalar functions (typically the quantum wavefunction of an electron in a conduction band), and then in a subsequent step expose the transformation laws in the spinorial case (typically a hole in the valence band). To eliminate any ambiguity in the following we shall always use a passive point of view for the symmetry operations, i.e. the operations are always considered as coordinate transformations linked with a change in reference frame, i.e. they are not rotations of the physical system.

iii.1.1 Coordinate transformations

Let us first define a basic transformation of coordinates linked with a change in orthonormal cartesian reference frame. It is defined by an orthogonal matrix belonging to and defining the basis vectors of the new frame with respect to the old one:


where . It is either a rotation or an improper rotation, any rotation can be parametrized by its Euler angles , and , and the corresponding set of rotation matrices define a representation of the rotation group (). These matrices can be systematically constructed using the generators of the rotations Messiah (1969):


where are the components of the angular momentum pseudo-vector . Improper rotations can always be decomposed as the product of the spatial inversion with a proper rotation, therefore their matrix representation is written as the product of with the corresponding as defined in Eq. (12. A typical example is , a mirror symmetry with normal , which is the product of the inversion with , a -rotation around the axis . Therefore , where temporarily the Euler angle notation is left out. Similarly, in the following we shall use to denote the change in coordinates linked with any arbitrary change of orthonormal reference frame, and often (without argument) will denote implicitely the corresponding orthogonal matrix representation (the argument of will be restored only when absolutely needed for clarity).

Let us now recall how the set of components of a vector with respect to a given basis transform under a change in coordinates (new basis ):


which is a contragredient law with respect to eq.(11), as it should for a passive point of view.

iii.1.2 Transformation of scalar functions

In the Hilbert space corresponding to the set of possible electronic wavefunctions in confined dimensions (d=1,2,3), one can associate linear operators to every possible coordinate transformation :


The new mathematical function is a function of the new coordinates, but is defined as representing the same quantum state, i.e.


It is easy to check that this definition leads to the expected multiplication rule for two successive coordinate transformations and .

Let us now consider the generic form of the scalar Hamiltonian in mixed position and momentum representation (of complementary dimensions and respectively) which appeared in Section II.1 (Eq. (1)). This generic form is applicable for heterostructures of any dimensionality , from 3D to 0D ( is the dimensionality in -space related to the “free-like” motion of charge carriers in the directions with full translationnal invariance at the heterostructure level). The arguments should thus be understood as follows:

The new transformed Hamiltonian operator is obtained by enforcing (in Dirac notation) , which leads to


Note that k is implicitely understood as the vectorial components of the wavevector (covector). In Eq. (17) the operations are always considered in 3D (i.e. belonging to ), therefore the eigenfunctions of the generic Hamiltonian must have an auxiliary index in order to transform consistently, i.e.


Such a generic scalar Hamiltonian is typically a quadratic form of the momentum, i.e. of the components of along the non-confined directions (translational invariance), and of differential operators which are kept in the confined directions (according to the correspondance ). The parameters of this Hamiltonian, i.e. the effective masses and the confinement potential, are functions of positions like in Eq. (1). It may also include anisotropic masses as follows: , where is the “matrix” (tensor) of coefficients and is a scalar operator. In this rather general case the new Hamiltonian is obtained by




Is easy to understand equation (20) when C is independent of r: is a scalar, invariant under a passive transformation and k are the vectorial components of wave vector, then C is a tensor two times covariant and, for this kind of tensors, the tranformation laws give .

iii.1.3 Transformation of spinors and spinorial fields

In a model including the spin of the charge carriers, the quantum state may generally be described by a spinorial field . In this section, for clarity, we will use the underscore to explicitely denote the spinorial character, which is assumed to be of dimension where is a strictly positive half-integer. In the multiband envelope function formalism, the spinorial components are related to the Bloch function basis at a high symmetry point of the Brillouin zone, diagonalizing the Hamiltonian of the charge carrier in the bulk crystal structure Bassani and Parravicini (1975).
The corresponding Hilbert space is spanned by the tensor products of spinors belonging to with the envelope functions belonging to . Hence when considering truly arbitrary coordinate transformations one should consider to perform the transformations independently in both spaces.
Let us first look solely to spinor transformations. By contrast to vectors in normal cartesian 3D space (c.f. Eq. (11)), spinors obey different transformation rules under rotations, called spinorial transformation rules Appel (2002); Carmeli and Malin (2000). For pure rotations the transformation rules can always be related by a similarity transformation to a set of matrices , indexed by the Euler angles, and called the Wigner representation Messiah (1969); Wigner (1971), which is a -dimensional projective representation of :


When is half-integer it is a spinor representation of (instead of ). The double underscore notation introduced here always denotes a square matrix character in spinorial space. A typical feature of spinor representations is that a rotation around any axis will always be associated to a sign change, i.e. . For the representation of improper rotations one can again use the factorization of the inversion, however a small complication is the Wigner representation of the inversion, which can be either . We shall not discuss here the proper choice of the latter sign since it will not matter within this paper. In some cases it might be important, see Dalessi et al. (2009).

A crucial feature of our approach is now to consider more general unitary change in coordinates , not linked with 3D rotations, characterized by the general set of square matrices of dimension belonging to the group .
Let us start with the -dimensional Bloch functions basis denoted , as is customary in the field, and a new basis differing by a rotation , so that:


which is similar to the standard transformation law of partner functions linked with the irrep of dimension of the rotation group under . It is important to make here two remarks for clarity. First we stress that for such Bloch functions simply correspond to a label related to the transformation law (22) and to the dimensionality of the Bloch functions basis, but in this context it is not a true angular momentum quantum number (it just refers to the transformation properties, and not truly to rotations). A second important remark is that the basis, and its corresponding transformation law under rotations (22), already incorporate, in addition to pure spin, an orbital part, since the “bulk” spin-orbit interaction is already diagonalized by the Bloch function basis. Later below a purely orbital transformation, operating solely on the envelope function, will have to be simultaneously added. When one considers more general transformations of coordinates in spinorial space ():


the spinor components can be transformed using the matrix equation


which is anologous to Eq. (13). The matrix defines a more general basis change than the one specifed by Eq. (22).
Let us now consider all the possible transformations of spinorial fields, which are defined by two separate coordinates transformations in both spaces in real space and and combine them in associated with acting in the tensor product Hilbert space:


where is the operator defined by Eq. (14) and by Eq. (23). Here we have kept the possibility of arbitrary and different changes of coordinates in both spaces, which will prove crucial later. Under a coordinate transformation an arbitrary quantum state of our system, described by a spinorial field, transforms therefore like


We see clearly in Eq. (26) that the spinorial character of the transformation does couple different envelope function components through the matrix, and is more complicated than the simple 3D transformation of the envelope functions appearing in Eq. (18). This complication is one of the roots of the difficulty in interpreting the individual symmetry of envelope functions in a heterostruture with a given symmetry.
In the spinorial case one can transform the Hamiltonian operator matrix too, but one must take into account the presence of the matrix in a similar way:


where, in the last two lines, we have used again the shorthand notation and .

iii.2 Symmetry and resulting constraints on eigenstate envelope functions

Let us now assume a heterostructure with a given symmetry group of spatial transformations. More precisely, if there is translationnal invariance (i.e. ), let us take as the small point group of of the structure, with cardinality , and defined by the restriction:


The operations are either pure rotations or roto-inversions (mirrors). The heart of the passive point of view is just to express than any full coordinate transformation corresponding to a symmetry element of the structure will leave invariant the form of the k-restriction of the conduction and valence band Hamiltonians (hence from now on we shall leave out the implicit k-subscript on eigenstates and envelope functions appearing in Eqs. (18) and (26).
For the conduction band the product of two symmetry operations and will simply follow the multiplication table of the single point group , whilst for the valence band one must necessarily use a double group notation of symmetry operations due to its spinorial nature (please note that here we use the notation to denote any element of , instead of the single group element multiplied by a -rotation, as is quite standard (see e.g. Altmann and Herzig (1994))). The composite index for the corresponding full coordinate transformation can be identified with , provided one understands it as where is the single point group image of . Clearly this composite index obeys also the double point group multiplication table since the product is defined as .
Using Eq. (27) let us now express the invariance of the k-restricted Hamiltonian with respect to a given symmetry operation of the coordinate system:


where shorthand notations have been used. Eq. (29) is equivalent to state that every symmetry operation commutes with the k-restricted Hamiltonian, which allows the use of a well known theorem Bassani and Parravicini (1975) that states that every eigenspace of the Hamiltonian can then be labeled by an irreducible representation (irrep) of (meaning that except for accidental degeneracies its dimension is necessarily the dimension of the irrep), and that a basis of partner eigenstates ( is called the partner function index) can be found such that under a basis change it transforms according to


where the set form a unitary irreducible matrix representation of the irrep .

From the physical point of view, Eq. (30) means that a transformed symmetrized eigenstate under does not only have the same energy but can also be developed on its partners, so that there are constraints on individually transformed envelope functions since they must “reconnect” on all the other ones in a very intricate fashion. Using Eq. (26) we find


This equation forms the starting point of our theory. It will be exploited in the next section where we use the explicit separation of the spatial and the spinorial part of the operators in Eq. (25), and then seeks the basis that will minimally reconnect envelope functions according to (31).

Iv The fully symmetrized OBB basis and the separation of spinorial and spatial parts

By definition, a 3D representation of any point group can be obtained simply from the analytical expression of the matrices (12) and by factoring out the inversion for improper operations. In spinorial space things are slightly more complicated since, depending on the basis used to express the Hamiltonian, the matrices do form a spinorial -dimensional representation of the double group , but not necessarily given by Wigner matrices (21).

In this section we shall start from a spinorial Bloch function basis which, at high symmetry points of the Brillouin zone, is often denoted , due to its symmmetry transformation properties. For the valence-band of semiconductors like , at the so-called -point, one restricts to  Bastard (1988) giving rise to the Luttinger Hamiltonian (c.f. Eqs. (3) and (4)). However the formalism holds for an arbitrary large number of bands. The heterostructure symmetry is kept general, but illustrated with symmetry. A few other cases, in particular the symmetry groups, will be shortly discussed in Sec. VIII.
To introduce the concept of the Optimal Bloch function Basis (OBB) with respect to a heterostructure with a given symmetry, we must heavily rely on the explicit separation of orbital and spinorial part carried out in the last section, and on the possibility of separate coordinate transformations in both spaces.

iv.1 The optimal Bloch function basis

Our main goal is to simplify Eq. (31) and minimize the coupling between different envelope functions. The main novel idea is to perform once for all a unitary coordinate transformation , i.e. purely in spinorial space, corresponding to a best choice of the Bloch function basis, such that the set of reducible matrices appearing in (31) would become block-diagonal!

Let us first note that some aspects of this idea are not completely new. Up to now one as used 3D rotations of the quantization axis (direction of ), parametrized by the Euler angles such that would be transformed towards , diagonalized by  Fishman (1995); Dupertuis et al. (2000). The corresponding image of is the so-called Optimal Quantization Axis (OQA) Dupertuis et al. (2000). The more trivial case of quantum wells grown in direction, where the OQA is always , is treated in Fishman (1995). The OQA was supposedly considered the best choice Dupertuis et al. (2000) to simplify the Luttinger Hamiltonian. Indeed a change in basis has lead to a new form of the Hamiltonian expressed in the new basis as


which lead to simpler wavefunctions in the case. It also lead to a corresponding new representation of the symmetry operations


The optimal choice of angles was carefully made Fishman (1995); Dupertuis et al. (2000), and discussed partly in Sec. II, but we have seen that it cannot be efficient for higher symmetries.

In the present approach, the novelty is to consider more general unitary transformations that cannot be represented as rotations of the original reference frame. For this purpose we shall seek the coordinate transformation , but possibly , towards new Bloch states , where is the label of a double group irrep, and a corresponding partner function label. This set of states, that we shall call Optimal Bloch function Basis (OBB), is a fully symmetrized Bloch basis, i.e. symmetrized according to the symmetry of the quantum heterostructure. From its definition


The notation makes it clear that every new basis state will map under passive symmetry operations like the standard set of partner functions of an irrep of the double group:


but with on the right hand side. Let us now look at the transformation properties of the new spinorial components of the field , transforming as


From Eq. (35) it is clear that the change in basis specified by the matrix induces automatically a new block-diagonal representation of the symmetry operations where Eq. (33) then becomes


The blocks of dimension of are labelled by and a possible multiplicity index runnig from to . Indeed a given representation may appear more than once (a pratical example is the group rediscussed later), however for simplicity we shall forget it since it can be restored without difficulty.
Let us now separate every irreducible blocks of , and of every partner eigenstate of the irrep :