Symmetry of electron states in semiconductor structures under a magnetic field

phys. stat. sol. (b) 244, No. 6, 2010–2021 (2007) / DOI 10.1002/pssb.200642446

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Symmetry of electron states in semiconductor structures

under a magnetic field

P. Tronc*, 1

and V. P. Smirnov2

1 Laboratoire d’Optique Physique, Ecole Supérieure de Physique et Chimie Industrielles,

10 rue Vauquelin, 75005 Paris, France 2 Department of Mathematics, Institute of Fine Mechanics and Optics, Sablinskaya ul. 14,

197101 St Petersburg, Russia

Received 25 August 2006, revised 5 December 2006, accepted 3 January 2007

Published online 28 February 2007

PACS 73.21.–b, 75.90.+w

We present a group-theory analysis of the electron states in bulk and low-dimensional semiconductor

structures under a uniform magnetic field. The analysis takes into account the gauge transformations un-

der the symmetry operations. It is shown that the symmetry operations commute in dots, rods and tubes

whatever is the orientation of the magnetic field whereas they commute in layers only when the field is in

the plane. The commutations properties allow using conventional symmetry groups. In bulk materials, the

confinement of electrons within rods whose axes are parallel to the field makes it possible to derive ap-

proximate wavefunction symmetries from the rod group involving the geometrical symmetry operations

common to the field and the crystal but excluding the translations that are not parallel to the field. An ap-

proximation of the same kind can also be made for low-dimensional structures. Applications to bulk mate-

rials with the wurtzite or zinc blende structure as well as related nanostructures are presented.

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

The effect of a uniform magnetic field applied to a semiconductor structure (bulk or nanostructure) has first been studied by modeling the material properties with the help of carrier effective masses only. Such a treatment provides the celebrated Landau levels. Besides, another approximation is commonly used that consists in calculating the coupling energy between the electron spin and the magnetic field with the help of the unperturbed wavefunctions (Zeeman splitting). Later, Koster et al. [1] proposed, through the Compatibility Tables, a method for the analysis of the 32 point groups under a magnetic field and it was pointed out by Wannier and Fredkin [2] that a uniform field physically does not destroy the transla-tional invariance of a structure since the physical environment of the electron is the same at all sites whose positions differ by a lattice vector. Brown noticed [3] that a type of translation operator should exist under which the Hamiltonian is invariant and defined a set of translation operators (the magnetic translation group). The operators form a ray group [3]. Ashby and Miller [4] and Zak [5, 6] studied the magnetic translation group in bulk materials and Zak [6] proposed a method to get its irreducible repre-sentations (IRs). To our knowledge, the formalism of magnetic translations has never been used to solve concrete problems except in the case of the Landau approximation for the study of quantum wells (QWs) [7]. That perhaps arises from the complexity of the theory but it should also be noticed that the theory does not consider the point symmetry operations kept under the magnetic field, namely the rotations

* Corresponding author: e-mail: [email protected], [email protected]

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(proper and improper) around the direction of the field, the symmetry or glide planes perpendicular to it and their products. As a consequence of the point symmetry operations, the group of the magnetic trans-lations in the plane defined by the direction of the magnetic field and one of the two other vectors of the associated unit cell is no longer commutative and its IRs are no longer 1-dimensional as they are in the magnetic translation group [5, 6] (see Section 2). It seems necessary to consider again the problem of the symmetry of the structures under a magnetic field since numerous inadequacies do exist in the Landau and Zeeman approximations that have not been resolved in the theory of magnetic translations. For example: i) The Landau model is based on a free-electron approximation. As a result, the component of the angular momentum along the magnetic field is a good quantum number. On the contrary, when taking into account the exact symmetry of a structure, only 2-, 3-, 4-, and 6-fold rotation axes can eventually exist and the component of the electron angular momentum cannot longer be a good quantum number. For orbitals with a small expansion in space, an isotropic approximation is clearly inadequate since the site symmetry becomes preponderant. This is the case for carriers and excitons tightly bound to impuri-ties and defects or tightly confined in nanostructures such as thin QWs or small quantum dots (QDs), for any exciton with a large Rydberg value, and for any carrier or exciton submitted to an intense magnetic field. ii) The magnetic field keeps a symmetry plane only if the latter is perpendicular to it. On the contrary, the Zeeman model keeps any symmetry plane since the model is based on the interaction of the magnetic field with the spin of the unperturbed wavefunctions that take into account the effect of any symmetry plane present in the structure. In the particular case of nanostructures, the Envelope Function Approxi-mation (EFA) artificially introduces symmetry operations that actually do not exist, such as a symmetry plane located at the center of each superlattice (SL) slab or at the center of QWs. The consequences of such artifacts on the symmetry of the system under a magnetic field and hence on the optical selection rules are not usually considered but should be carefully analyzed within a more accurate theory. Due to the lack of knowledge about the symmetry of the system under the magnetic field, the optical selection rules cannot be safely derived. Generally, the angular-momentum-conservation law is used to get selection rules between approximate wavefunctions but discrepancies are present at any finite value of the magnetic field strength due to the change of symmetry. They increase with the magnetic field strength or, in other words, with the energy associated to the Bohr magneton (one can now achieve fields of the order of 50 T). Last, it should be noticed that, contrary to the case of semiconductor structures under a magnetic field for which only approximate models are used, the full symmetry has been determined for magnetic mate-rials with the help of Shubnikov (color) space groups [8].

2 General considerations

A uniform magnetic field B is invariant under the operations of the F symmetry group that involves the proper and improper rotations around axes parallel to B, the symmetry planes perpendicular to B, any translation, and their products. Let the vector potential be described with the help of the symmetric gauge:

1

2[ ] .= ¥A B r (1)

It is well known that the Schrödinger equation for an electron (mass m and charge –e) in the crystal potential V(r) submitted to the magnetic field B can be written as:

, ,

( ) ( ) ,j n j j nH EΨ Ψ=r r

2

1ˆ( ) ,

2

e eH V

m c mc

Ê ˆ= + + + ◊Ë ¯p A r s B (2)

2012 P. Tronc and V. P. Smirnov: Symmetry of electron states in semiconductor structures

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.pss-b.com

where i�= - —p is the momentum operator, s the spin operator, j the eigenvalue index and n enumerates the eigenstates with the same energy. Let us submit Eq. (2) to the transformation 1 1( )g R

- -

Æ = -r r r a

where ( )g R= a is the rotation R followed by the translation a. If g is an element of the symmetry group G (c) of the crystal V(g–1

r) = V(r). Under this operation, 1R

-

Æp p, 1( ) (1/2) [ ( )]R-

Æ ¥ -A r B r a

1 ( )R-

= -A r a , the term ˆ ◊s B does not change since it does not depend on r. The scalar product of vec-tors depending not on the orthogonal operation R–1 applied to both vectors, 1 2( ( ( / ) ( )))R e c

-

+ -p A r a 2( ( / ) ( ))e c= + -p A r a , Eq. (2) becomes:

2

1 1

, ,

1ˆ[ ( )] ( ) ( ) ( ( )) .

2 2j j j

e eV g E R

m c mcµ µ

Ψ Ψ- -

Ê ˆÊ ˆ+ ¥ - + + ◊ = -Á ˜Ë ¯Ë ¯p B r a r s B r r a (3)

As it is well known, the gauge transformation A(r) → A(r) + ( )f— r induces the following change of the wave function in the Schrödinger Eq. (2):

( ) exp ( ) ( ) .e

i fc

Ψ Ψ�

Ê ˆÆ Ë ¯r r r (4)

Supposing ( ) (1/2) [ ]f = - ¥ ◊r B a r , one finds A(r) → (1/2) [ ( )]¥ -B r a , and, under this gauge trans-formation, Eq. (2) takes the form:

2

,

,

1ˆ( [ ( )]) ( ) exp [ ] ( )

2 2 2

exp [ ] ( ) .2

j

j j

e e eV i

m c mc c

eE i

c

µ

µ

Ψ

Ψ

�

�

Ê ˆ Ê ˆ+ ¥ - + + ◊ - ¥ ◊Ë ¯ Ë ¯

Ê ˆ= - ¥ ◊Ë ¯

p B r a r s B B a r r

B a r r

In other words, the change of the vector potential A(r) under geometrical transformation 1g

-

Ær r can be compensated by the gauge transformation with the function:

1

2( ) ([ ] ) .f = ¥ ◊r B a r (6)

Note that for a = 0 or a || B the function f(r) is equal to zero and A(r) does not change at all. Equations (3) and (5) being identical, their solutions corresponding to the same energy level Ej are linear combina-tions ones of the others

1

, ,( ) exp [ ] ( ) ( ) .2

j j

eg i C g

cµ µ µ µ

µ

Ψ Ψ�

-

¢ ¢

¢

Ê ˆ= - ¥ ◊Ë ¯Âr B a r r (7)

Therefore the symmetry elements of H in Eq. (2) are those elements ( )g R= a of the group G(c) for which RB = B, this operation being followed by the appropriate gauge transformation with the function (6). Hereafter symmetry elements of the Hamiltonian H are denoted by g*. These elements form the group G* isomorphic to the group G ⊂ G(c) of purely geometrical transformations g. The latter is the intersection of the groups G(c) and F. It is always possible to choose the direction of the magnetic field to be the direction of the a3 primitive lattice vector. The elements g* ∈ G* with a = 0 or a || B are pure geometrical symmetry elements ( f(r) = 0, g* = g) and form for bulk crystals the 3-dimensional 1-periodic rod group GRod ⊂ G*. The group G* contains also the subgroup of magnetic translations

M*T with the elements

1 1 2 2( )*E n n+a a where n1, n2 are

integers and a1, a2 are primitive vectors. As the subgroups GRod and M*T have no common elements except

the identity and M*T is an invariant subgroup in G*, the group can be represented as the semi-direct prod-

uct:

M Rod** .G T G= Ÿ (8)

(5)

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Basic functions in the space of an IR of G* transform according to a law that differs from the trans-formation law of usual IRs:

* 1

, , ,( ) exp [ ] ( ) ( ) ( ) ,2

j j j

eg i g C g

cµ µ µ µ µ

µ

Ψ Ψ Ψ�

-

¢ ¢

¢

Ê ˆ= ¥ ◊ =Ë ¯ Âr B a r r r (9)

a being a proper or improper lattice translations. A consecutive application of the symmetry elements 1g*

and 2

g* to the wave functions ( ),j µΨ r

1

2 1 , 2 2 1 , 1

1 1 1

1 2 2 2 , 1 2

1

2 2 1 2 1 2 , 2 1

2 1 ,

( ) ( | )*exp [ ] ( )2

exp {[ ] ( ) [ ] } ( )2

exp [ ( )] exp [ ] (( ) )2 2

[ ( ) ( )]

j j

j

j

j

eg g R i g

c

ei R g g

c

e ei R i R g g

c c

C g C g

µ µ

µ

µ

µ µ

µ

Ψ Ψ

Ψ

Ψ

Ψ

�

�

� �

-

- - -

-

¢¢

¢¢

Ê ˆ* *◊ = ¥ ◊Ë ¯

Ê ˆ= ¥ ◊ - + ¥ ◊Ë ¯

Ê ˆ Ê ˆ= ¥ + ◊ ¥ ◊Ë ¯ Ë ¯

=Â

r a B a r r

B a r a B a r r

B a a r a a B r

( )µ ¢¢

r

does not give the same result as the product 3 2 1 2 1 2 2 1

*( | )g g g R R R* * *= = +a a :

1

2 1 , 2 2 1 , 2 1

2 1 ,

( ) exp [ ( ] (( ) )2

( ) ( ) .

j j

j

eg g i R g g

c

C g g

µ µ

µ µ µ

µ

Ψ Ψ

Ψ

�

-

¢¢ ¢¢

¢¢

Ê ˆ* * = ¥ + ◊Ë ¯

=Â

r B a a r r

r

Comparing (10) and (11), one finds the multiplication law of transformation matrices ( )C g

2 1 2 1 3

( , ) ,g g g g gω* * *= 2 1 2 1 2 1

( ) ( ) ( , ) ( ) ,C g C g g g C g gω= (12)

where the factor system

2 1 2 2 1

( , ) exp [ ] .2

eg g i R

cω

�

Ê ˆ= - ◊ ¥Ë ¯B a a (13)

Since 2 1

| ( , ) | 1g gω = and

3 2 3 2 1 3 2 1 2 1

( , ) ( , ) ( , ) ( , ) ,g g g g g g g g g gω ω ω ω= (14)

the representations of the group of matrices C(g) are projective [3]. Note that in the particular case of pure translations (R1 = R2 = E), one has:

2 1 2 1 2 1

( , ) ( , ) exp [ ] ( , )2

eC E C E i C E

c�

Ê ˆ= - ◊ ¥ +Ë ¯a a B a a a a

and

1 2 1 2 1 2

( , ) ( , ) exp [ ] ( , ) ,2

eC E C E i C E

c�

Ê ˆ= - ◊ ¥ +Ë ¯a a B a a a a (15)

i.e. transformation matrices for translations ( , )C E a do not commute in the general case, and projective IRs are not 1-dimensional ones. When at least one of the translations is parallel to B, or when both trans-lations are in the same direction, or when B, a1, and a2 are coplanar then

1 2exp ( ( /2 ) [ ])i e c�- ◊ ¥B a a = 1,

and the commutation takes place as usual for translations multiplication law. To compare our results with those from the magnetic translation group, one can consider two symme-try operations

1g* and

2g* whose lattice translations a1 and a2 are in the (a1, a3) plane of the unit cell of a

(10)

(11)



bulk crystal. It can be readily shown that ω(g1, g2) and ω(g2, g1) in (13) are in general different one from the other, contrarly to the case of pure lattice translations. Indeed, the difference between

2 2 1[ ]R◊ ¥B a a

and 1 1 2

[ ]R◊ ¥B a a is the same as between 2 2 1

[ ]R◊ ¥¢ ¢B a a and 1 1 2

[ ]R◊ ¥¢ ¢B a a , the primes indicating the components of the a1 and a2 translations along the direction perpendicular to the magnetic field. The commutation takes place (ω(g1, g2) = ω(g2, g1) = 1) only when the angles of the R1 and R2 rotations are both equal to zero or π. The magnetic translation group corresponds to the case when both angles are equal to zero. The commutation takes also place for some couples of rotations around a 4- or 6-fold axis, namely when the angles of the R1 and R2 rotations differ by π, but ω(g1, g2) = ω(g2, g1) is no longer equal to 1. Therefore, in the general case, the two symmetry operations

1g* and

2g* with lattice translations a1

and a2 in the (a1, a3) plane do not commute and the procedure proposed in the theory of magnetic transla-tions [6] cannot be used to obtain the IRs of the Hamiltonian symmetry group. Taking into account the time reversal symmetry one obtains cogroups GT = G + θC2xG where θ = σ

yK is the time reversal operator (σ

y is the Pauli matrix and K is the complex conjugation operator).

As it is well known, the time reversal symmetry does not imply a new classification for energy levels since irreducible corepresentations of cogroups are unambiguously determined by the IRs of their unitary subgroups. Therefore later on, one does not consider explicitly antiunitary time reversal symmetry ele-ments.

3 Application to various types of structure

The G(c) group is a space group for a bulk crystal or a SL made of stochiometric compounds such as (GaN)

m(AlN)

nSLs for example. Indeed, these SLs have perfectly defined structures and can be rigorously

studied from the point of view of symmetry. Under the same conditions of stochiometric compounds, the symmetry of V(r) is described by layer groups for QWs, line groups (sometimes rod groups) for rods and tubes, and point groups for QDs and molecules, respectively. The symmetry of QDs depends on their shape. 1. Since a = 0 for dots and molecules, the function f(r) in (6) is equal to zero, and one deals with the usual representations of the group G. That also occurs for rods or tubes whatever is the direction of the magnetic field, since a2 and R2a1 (see Eq. (13)) are necessarly parallel one to the other and to the axis of the structure. That also holds for layers and QWs but only when the magnetic field lies within the plane since B, a1, and a2 are coplanar. 2. For bulk crystals, one can take advantage ot the electron – wavefunction confinement when the magnetic field is applied. It allows determining approximate symmetries for the wavefunction. Indeed, It has been widely verified in experiments dealing with a semiconductor structure submitted to a magnetic field that electron orbitals can be considered with a good accuracy as confined to restricted areas in the direction of the plane perpendicular to the field, the larger the field, the smaller the areas. Such an effect has been calculated in the Landau approximation and appeared in numerous textbooks (see for example Ref. [9]). The approximate symmetries of the states are then described by the IRs of the GRod group since the translational symmetry is left in the (a1, a2) plane. The eigenstates and eigenenergies are determined regardless to the magnetic translations. That can be shown in an other way. Indeed, due to the confine-ment, it seems sufficient for describing the wavefunction to consider an area in the (a1, a2) plane that should be large when compared to the wavefunction extension. Within the area, the vector potential can be writen as:

1 2

1 2

1 1 2 2

1 1 2 2

1( ) ( )

2(2 1) (2 1)

1 1 1 1,

2 (2 1) (2 1) 2

x y

x y

x y

x y

n n

m n m nx y

n n

m n m nx y

m m

n n

m m

n n

=- =-

=- =-

= ¥ - -

+ +

= ¥ - ¥ - ¥ = ¥

+ +

Â Â

Â Â

A r B r

B r B B B r

a a

a a

where nx and n

y are integers. When using this approximation, the vector potential and the Hamiltonian

have translational invariance in any direction and the wavefunction deduced from a wavefunction Ψ(r)

(16)

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by a primitive translation keeps the same energy. It is therefore sufficient to solve the Schroedinger equa-tion for the Grod group with the help of the conventional symmetric gauge. The effect of the magnetic translations in the (a1, a2) plane therefore reduces to that of simple geometric translations that keep the eigenenergy. One can notice that the result is different for that obtained with an electric field. Indeed, in the latter case, the eigenenergy is changed by an amount equal to eEa where E is the electric field strength and a the translation (see for example the Wannier–Stark effect in SLs submitted to an electric field parallel to the growth direction). 3. The same kind of approximation can be made for QWs when the field is not in the layer plane. The wavefunction is confined in the three directions and the opproximate symmetry is described by a point group. Hereafter the labeling of point group IRs follows Ref. [8]. The translation subgroup of the rod group consists in the lattice translations in the direction of the magnetic field for bulk materials and SLs whatever is the direction of the magnetic field. That also holds for QWs when the field lies within the plane of the layers. For rods or tubes the translation subgroup consists in the rod and tube lattice translations whatever is the orientation of the field. We do not con-sider hereafter rods or tubes whose space symmetry is described by line groups (for example carbon nanotubes) and limit ourselves to rods and tubes whose space symmetry is described by rod groups. The rod groups are subgroups of 3-periodic 3-dimensional space groups. For the rod groups we use the notations of Ref. [10] with the z-axis being parallel to the magnetic field. Beside the translations, the only possible symmetry elements in the rod groups are 2-, 3-, 4-, and 6-fold pure or screw axes, symme-try planes perpendicular to them, inversion, and S2,4,6 axes. Each rod group has the same symmetry opera-tions, except the translations along the x- and y-axes, as one particular 3-dimensional 3-periodic space group that will be labeled hereafter as the corresponding group [11]. The 1-dimensional BZ of the rod group coincides with the k

z restriction of the BZ of the corresponding 3-dimensional 3-periodic space

group and presents the same symmetry properties as the restriction. Therefore the IRs of the rod groups can be taken directly from the tables of the IRs of the corresponding 3-dimensional 3-periodic space groups. The rod group involves any point symmetry element of the structure that is not lifted by the field (the magnetic field is an axial vector). The optical selection rules can be established using the conventional procedure. In particular, the wavevector has to be kept in a direct transition. Within the 1-dimensional BZ, the optical selection rules are the same as for the corresponding 3-dimensional 3-periodic space group along the k

z-axis.

Sections 4 and 5 hereafter provide the results of our model for bulk materials with the wurtzite or zinc blende structure as well as for related nanostructures such as superlattices, quantum wells, and quantum dots.

4 Bulk semiconductors with the wurtzite structure

and related nanostructures

The bulk materials with the wurtzite structure and the SLs of the (GaN)m(AlN)

n type with odd values of

m + n have the C 46v non-symmorphic space group [12, 13]. Their C6v point group arises from a 3-fold

rotation axis combined with a π/3 and half-lattice-parameter improper rotation along an axis (63 screw axis) parallel to the former. There are also three mirror planes and three glide planes parallel to the sym-metry axis. For these structures one has the following results: 1. When the magnetic field is directed along the symmetry axis, the symmetry group of the structure is the rod group R 56 (p63). It contains a 63 screw axis and the lattice translations along the symmetry axis. 2. When the magnetic field is directed perpendicular to a mirror plane, the symmetry group of the structure is the rod group R 10 (p11m). It contains the reflection in the plane and the lattice translations along the magnetic field. 3. When the magnetic field is directed perpendicular to a glide plane, the group is the rod group R 1 (p1). It contains only the lattice translations along the magnetic field.



Table 1 Symmetry of bulk semiconductors with the wurtzite structure and related nanostructures under

a magnetic field. The notations σ and σ′ refer to symmetry planes and glide planes, respectively.

bulk material, SL (C46v) B || C6 R 56 (p63)

B^σ R 10 (p11m) B^σ ¢ R 1 (p1)

SL (C13v) B || C6 R 42 (p3)

B^σ R 10 (p11m)

QW (DG 69) B || C3 C3

B^σ R 10 (p11m)

QD (C3v) B || C3 C3

B^σ Cs

For the SLs with an even value of m + n (they have the C 13v symmorphic space group [12, 13]) one has

the following results: 1. When the magnetic field is directed along the symmetry axis, the group is the rod group R 42 (p3). It consists in the C3 axis and the lattice translations along the axis. 2. When the magnetic field is directed perpendicular to a mirror plane, the group is the rod group R 10 (p11m). It contains the reflection in the plane and the lattice translations along the magnetic field. For the QWs (their space symmetry is described by the layer group L 69 (p3m1) [12]) one has: 1. When the magnetic field is directed along the symmetry axis, the group is the point group C3. 2. When the magnetic field is directed perpendicular to a mirror plane, the group is the rod group R 10 (p11m). It contains the reflection in the plane and the lattice translations along the direction of the magnetic field. For the QDs (their symmetry is described [14] by the C3v point group) one has: 1. When the magnetic field is directed along the symmetry axis, the group is the point group C3. 2. When the magnetic field is directed perpendicular to a mirror plane, the group is the point group Cs. The rods and tubes have been considered in an other paper [15]. All the results are displayed in Table 1. In addition, Table 2 provides the correspondence between the rod groups mentioned in this section and in the following one, with the 3-dimensional 3-periodic space groups.

5 Bulk semiconductors with the zinc blende structure

and related nanostructures

In the same manner as in Section 4, one can study the zinc-blende-based structures (the results of the present section are summarized in Table 3).

Table 2 Correspondence between rod and space groups.

rod group corresponding space group

R 1 (p1) 1 (C11)

R 3 (p211) 3 (C21)

R 8 (p112) 3 (C21)

R 10 (p11m) 6 (Cs1)

R 27 (p4) 81 (S41)

R 42 (p3) 143 (C31)

R 56 (p63) 173 (C66)

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Table 3 Symmetry of bulk semiconductors with the zinc blende structure and related nanostructures un-

der a magnetic field. The notations S4, σ, C3, and U refer to S4 axes, symmetry planes, 3-fold symmetry

axes, and 2-fold symmetry axes, respectively.

field orientation structure symmetry

bulk materials, Td2 B || S4 R 27 (p4)

B^σ R 10 (p11m) B || C3 R 42 (p3) SL, D5

2d B || S4 R 27 (p4) B || U R 8 (p112) B^σ R 10 (p11m) SL, D9

2d B || S4 R 27 (p4) B || U R 8 (p112) B^σ R 10 (p11m) QW, L 59 B || S4 S4 B || U R 8 (p112) B^σ R 10 (p11m) QD, D2d B || S4 S4 B || U C2 B^σ Cs

The bulk materials with the zincblende structure, such as the GaAs or AlAs crystals for example, have the Td

2 symmorphic space group. The Td point group contains the covering operations of a regular tetra-hedron [16]. For the materials, one has the following results: 1. When the magnetic field is directed perpendicular to a cube face, the symmetry group of the struc-ture is the rod group R 27 (p4) consisting of the S4 axis along the direction of the magnetic field and of the lattice translations in the same direction. 2. When the magnetic field is directed perpendicular to one of the six diagonal reflection planes, the group of the system is the rod group R 10 (p11m). 3. When the magnetic field is directed along a 3-fold axis, the group of the system is the rod group R 42 (p3). 4. In the other cases, there is no point symmetry element except identity. The group is the rod group R 1 (p1). The SLs of the (GaAs)

m(AlAs)

n type grown along the [001] direction have the D5

2d or D92d symmorphic

space symmetry depending m + n is even or odd [17]. They have a S4 symmetry axis parallel to the growth direction, two perpendicular symmetry planes containing the S4 axis, and two 2-fold axes perpen-dicular to the S4 axis. The symmetries of the systems are given in Table 3 for different orientations of the magnetic field with respect to the crystal symmetry elements. In the other cases, there is no symmetry element except the translations along the direction of the magnetic field. The QWs of the (GaAs)

m/AlAs type grown along the [001] direction have the D2d point symmetry and

the L 59 (p4m2) space symmetry [18]. The translational symmetry is lifted along the growth direction. One has the following results: 1. When the magnetic field is directed parallel to the S4 symmetry axis, i.e., parallel to the growth direction, the symmetry of the structure is reduced to the point group S4. 2. When the magnetic field is directed parallel to a 2-fold rotation axis, the axis is kept. The symme-try group is the rod group R 8 (p112). 3. When the magnetic field is directed perpendicular to a symmetry plane (it lies within the plane of the layer), the plane is kept. The symmetry group is the rod group R 10 (p11m). 4. In the other cases, there is no symmetry element except the translations in the cases when the mag-netic field lies within the plane of the layers.



The square QDs of the GaAs/InAs type with a tetragonal shape have the symmetry described by the point group D2d. One get the following symmetries: S4 for B parallel to the S4-axis, C2 for B parallel to a 2-fold axis and Cs for B perpendicular to a symmetry plane.

6 Discussion

To illustrate our analysis of the electron states in semiconductor structures under a magnetic field, we consider hereafter few examples involving bulk materials and nanostructures based on the wurtzite or zinc blende lattice.

6.1 GaN/AlN quantum dots

The reduction from the C6v symmetry used for example in EFA for GaN/AlN QDs to the exact C3v sym-metry [19] reflects in the symmetry of the dots when a uniform magnetic field is applied. Table 4 pro-vides the symmetry of the C3v and C6v dots for various directions of the magnetic field. Differences be-tween both types of dots appear with the magnetic field being either parallel to the c (growth) axis or perpendicular to a σ′ symmetry plane (that does exist only within the C6v group). As a consequence, dif-ferences also appear in the symmetry of the wavefunctions and vibration modes and, hence, in the optical selection rules. With a magnetic field perpendicular to a σ′ plane, any transition is allowed in any polari-zation in the C3v dots since their symmetry is reduced to C1 whereas it is allowed in the polarization ei-ther parallel or perpendicular to the σ′ plane between states with identical or opposite spin orientations, respectively, in the C6v dots [19]. Features are also different for the C3v and C6v dots when the magnetic field is parallel to the c-axis [19] since their symmetry is described by the C3 and the C6 point group, respectively (Table 4). Of course, the specific effects arising from the exact C3v symmetry when compared to the approximate C6v one should be stronger for small dots than for large ones and should take place almost at high strength values of the magnetic field.

6.2 Quantum wells with the zinc blende structure grown along the [001] direction

The point symmetry of the (GaAs)m/AlAs QWs is described by the D2d point group whatever is the m

value. When applying a magnetic field parallel to the growth axis, the translational symmetry of the system in the layer plane is removed and its symmetry is reduced to the S4 point symmetry. The carrier symmetry is described by the ē1

(1), ē1(2), ē2

(1), and ē2(2)

double-valued IRs of the S4 group. Any single- or double-valued IR of the S4 group is unidimensional. The ē1

(1) and ē1(2), and ē2

(1) and ē2(2) IRs are complex

conjugate (corepresentations). Within each pair of corepresentations, the two IRs correspond to opposite spin values. Table 5 displays the Kronecker products of the IRs, providing the optical selection rules for fermions. Indeed, the vector representation in the S4 group is a (z) + e(2)(x – iy) + e(1)(x + iy). The a and b single-valued IRs are real whereas the e(1) and e(2) ones are corepresentations. When the a, e(2), or e(1) IR

Table 4 Symmetry of the C3v and C6v dots under a magnetic field. The notations σ and σ′ refer to the

symmetry planes existing in any dot and in the C6v dots only, respectively.

without the field with the field

B || c C3v C3 C6v C6 B ^ σ C3v Cs C6v Cs B ^ σ′ C3v C1 C6v Cs

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Table 5 Kronecker products of the double-valued IRs of the S4 point group and polarization in paren-

theses for the allowed transitions

ē1(1) ē2

(2) ē1(2) ē2

(1)

ē1(1) a(z) b e(2)(σ+) e(1)(σ –)

ē2(2) b a(z) e(1)(σ –) e(2)(σ+)

ē1(2) e(1)(σ –) e(2)(σ+) a(z) b

ē2(1) e(2)(σ+) e(1)(σ –) b a(z)

appears in the Kronecker product, the transition is allowed in the z, σ+ or σ – polarization. The transitions between two states with the same symmetry is allowed in the z polarization whereas the transitions be-tween two states with symmetries different one from the other are forbidden for states with the same spin value and allowed in a circularly polarization for states with opposite spin values. Kusrayev et al. [20] studied the linear polarization of the light in photoluminescence (PL) spectra at the Γ point of CdTe/Cd1-x Mn

xTe QWs with an in-plane magnetic field. They measured the linear polari-

zation degree ρ0 for the directions parallel and perpendicular to the magnetic field and the ρ45 degree for the directions rotated relative to the former ones by π/4 around the growth axis. For any well, the meas-ured linear polarization degree ρ45 vanishes for φ = υπ/2, where φ is the angle between the in-plane mag-netic field and the [110] crystal axis and υ is an integer. Both ρ0 and ρ45 degrees are periodic functions of φ with a period equal to π. Even though the barrier material is an alloy, i.e., a material without a perfectly defined symmetry, it is reasonable to assume that the symmetry properties established for GaAs/AlAs QWs still hold [16, 18, 21]. Indeed, the carrier presence probability is much lower in the barriers than it is in the wells. As a consequence, the influence of the barriers on the symmetry properties of the electron wavefunctions should be weak. If the PL spectra arise from transitions between band states only, to explain the properties of ρ0 and ρ45 mentionned above one can refer to the D2d symmetry that is the symmetry of unstrained QWs. In the symmetry, the two symmetry planes, as well as the two 2-fold symmetry axes, are not equivalent one to the other and the π period results from the D2d symmetry. When the magnetic field is applied perpendicu-lar to a symmetry plane, the plane is kept. Two orthogonal faces of the cube parallel to the growth direc-tion are deduced one from the other by the symmetry with respect to the plane. Hence ρ45 should vanish, that actually occurs. When the magnetic field is applied parallel to a 2-fold rotation axis, the axis is kept and two directions in the layer plane symmetric one from the other with respect to the direction of the magnetic field are equivalent. Therefore, the ρ45 degree should also vanish, that occurs only for the thick (60 Å wide) well. For the other wells (40 Å and 20 Å wide, respectively), one can therefore assume that the 2-fold axes are removed. The symmetry of the well is then lowered to C2v.

The π period is kept by the C2v symmetry. If the photoluminescence arises from transitions between a bound state and a band or a bound state (the bound state(s) being related to a substitutional impurity) the conclusion is different. Indeed, it as been shown [18] that at most two atomic sites within the QW unit cell have the D2d symmetry, the others having the C2v symmetry, with the two symmetry planes of any atomic site being parallel to those of the well structure. The atoms occupying the latter sites are therefore preponderant in the PL spectra and it is not necessary to assume a lowering of the structure symmetry from D2d to C2v to account for the experi-mental results concerning the 40 Å and 20 Å wide wells. After Kusrayev et al. [20], most of the intensity of the PL spectra of the wide well arises from D 0X transitions. They provided no data for the other wells, but if the origin of the PL spectra is the same, it is not necessary to assume a lowering of the D2d symmetry. The interpretation of the data for the wide well is more complex. Indeed, on one hand, the experimental results can be fitted by the D2d symmetry, but, on the other hand, the preponderance of D0X transitions in the PL spectra imposes that the dependency of the spectra versus the angle φ should be governed by the C2v symmetry. The fit of the experimental re-



sults with the D2d symmetry arises from the existence of nodes of the ρ0 degree for φ = π/4 + υ′π/2 where υ′ is an integer. Therefore, it should be concluded that these nodes cannot result from symmetry proper-ties of the structure. Finally, note that the possible symmetries of the Γ Bloch states with a magnetic field perpendicular to a symmetry plane, hence the corresponding optical selection rules, are not modified by a possible lower-ing of the structure symmetry from D2d to C2v since the Cs group is a subgroup of the C2v group that, in turn, is a subgroup of the D2d group.

6.3 Bulk materials

Bulk GaN has the C46v space symmetry. When applying a magnetic field parallel to the C6 axis, the sym-

metry is reduced to the non-symmorphic rod group R 56 (p63) (Table 3). The point symmetry is C6. The corresponding 3-dimensional triperiodic space group is C6

6 that shows that any point of the 1-dimensional BZ of the R 56 rod group has the C6 symmetry. The corresponding selection rules for the direct optical transitions can be found elsewhere [19]. Phonons are not sensitive to the magnetic field. Their symmetry therefore remains described by single-valued IRs of the C4

6v group. To study phonon-assisted optical

transitions, one has to subduce the IRs of the C46v

group onto its C66 subgroup. Any of the six single-valued

IRs of the C6 group can describe the symmetry of a phonon at every point of the reduced BZ [19]. The conventional procedure can be used to derive the selection rules. The Froelich interaction corresponds to the totally symmetric IRs of the C6 group [19]. Bulk GaAs has the Td

2 space symmetry. When applying a magnetic field parallel to an edge of the cube, the symmetry is reduced to the rod group R 27 (p4) (Table 3). The point symmetry is S4. The corre-sponding 3-dimensional triperiodic space group is S4

1 that shows that the Γ point and the point at the surface of the 1-dimensional BZ of the R 27 rod group has the S4 symmetry and the selection rules dis-played in Table 5 for the direct optical transitions. On the contrary, the other points in the reduced BZ have the C2 symmetry. Any direct optical transition is then allowed either in the z polarization (between states with symmetries different one from the other) or in the x, y one (between states with the same symmetry).

7 Conclusion

In few semiconductor structures (dots, molecules, rods, and tubes whatever is the orientation of the mag-netic field and QWs with the field in the plane), the symmetry operations of the electron Hamiltonian are commuting and one deals with conventional symmetry groups. In the other structures, the theory of magnetic translations is not sufficient to determine their symmetry when some point symmetry elements are not lifted by the field. Nevertheless, in such cases, the field-induced confinement of the electron wavefunction allows deriving approximate symmetries. The symmetries are described by rod or point groups.

Acknowledgement We acknowledge the PST.CLG. 979035 NATO Grant.

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Symmetry of electron states in semiconductor structures under a magnetic field