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© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Phys. Status Solidi B 246, No. 6, 1248 – 1251 (2009) / DOI 10.1002/pssb.200844212 p s s basic solid state physics b status solidi www.pss-b.com physica Exact space symmetry of electron states in lattices under a magnetic field P. Tronc * Laboratoire d’Optique Physique, Ecole Supérieure de Physique et Chimie Industrielles, 10 rue Vauquelin, 75005 Paris, France Received 21 May 2008, revised 7 November 2008, accepted 8 January 2009 Published online 16 February 2009 PACS 75.90.+ w, 78.07.–n, 78.20.– e * e-mail [email protected] © 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction Numerous authors [1 – 7] studied the problem of electron-Hamiltonian symmetry in three- periodic lattices under a uniform magnetic field B using the formalism of magnetic translations. Among the lattice symmetry operations, they took into account the lattice translations only, thus neglecting the point-symmetry op- erations except identity and inversion. In addition, they did not consider the reduction of the structure symmetry aris- ing from the magnetic field. As a result, their study actu- ally dealt with triclinic lattices only. Taking into account the reduction of symmetry shows that the symmetry of a given lattice is reduced to the triclinic one when the orien- tation of the magnetic field with respect to the lattice lifts any point-symmetry element except identity, and inversion if the latter is present in the lattice symmetry group. Never- theless, in many experiments the magnetic field is aligned with a high-symmetry direction of the lattice that allows in general keeping some other point-symmetry operations when the magnetic field is applied. For example, the zinc blende lattice has three orthogonal S 4 axis. Performing a complete symmetry analysis with taking into account the reduction of the lattice symmetry under the field [8] allows showing that when the field is parallel to one of the S 4 axis, the axis keeps its symmetry. The same result also holds for quantum wells (QWs) grown along a S 4 direction with the field being parallel to the direction: the S 4 axis perpendicu- lar to the layer plane is kept. In the wurtzite lattice, the c-axis has the C 6v symmetry that transforms into the C 6 one under a field parallel to the axis. Wurtzite QWs grown along the c-direction have a C 3v axis perpendicular to the layer plane that transforms into a C 3 axis under a field par- allel to the c-direction. Therefore, the formalism of mag- netic translations does not allow in the general case taking into account the effect of point-symmetry operations [8]. To my 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 QWs [6]. That perhaps arises from the weaknesses of the formal- ism that have been mentioned above as well as from its complexity. In addition, the formalism has been shown in Refs. [1 – 7] to provide three-periodic electron wavefunc- tions. Indeed, under their restricted conditions, the authors of Refs. [1 – 7] showed that it is possible to keep the full translational symmetry of the electron Hamiltonian and found such eigenfunctions. Besides, Trellakis [9] consid- ered a singular gauge transformation based on a lattice of magnetic flux lines, Cai et al. [10] proposed a formulation of electronic structure calculation based on a plane-wave basis and Fourrier transform approximation, and Nanciu [11] dealt with Wannier functions. The three authors pro- We previously established the transformations (including the gauge transformations) induced within the electron Hamilto- nian by the point- and space-symmetry operations of three- and low-periodic lattices under a uniform magnetic field [P. Tronc and V. P. Smirnov, Phys. Status Solidi B 244, 2010 (2007)]. We also determined the full symmetry group of elec- tron eigenfunctions by taking into account the electron con- finement in tubes whose axes are parallel to the magnetic field direction. The confinement is known from experiments with semiconductors. From the symmetry point of view, the confinement implies a lack of translational symmetry in the directions that are not parallel to the field. A theoretical proof is given here for the lack of translational symmetry in any lat- tice and the relation between the electron-Hamiltonian sym- metry group and the eigenfunction symmetry group is pro- vided.

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Page 1: Exact space symmetry of electron states in lattices under a magnetic field

© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Phys. Status Solidi B 246, No. 6, 1248–1251 (2009) / DOI 10.1002/pssb.200844212 p s sbasic solid state physics

b

statu

s

soli

di

www.pss-b.comph

ysi

ca

Exact space symmetry of electron states in lattices under a magnetic field

P. Tronc*

Laboratoire d’Optique Physique, Ecole Supérieure de Physique et Chimie Industrielles, 10 rue Vauquelin, 75005 Paris, France

Received 21 May 2008, revised 7 November 2008, accepted 8 January 2009

Published online 16 February 2009

PACS 75.90.+w, 78.07.–n, 78.20.–e

* e-mail [email protected]

© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction Numerous authors [1–7] studied the problem of electron-Hamiltonian symmetry in three-periodic lattices under a uniform magnetic field B using the formalism of magnetic translations. Among the lattice symmetry operations, they took into account the lattice translations only, thus neglecting the point-symmetry op-erations except identity and inversion. In addition, they did not consider the reduction of the structure symmetry aris-ing from the magnetic field. As a result, their study actu-ally dealt with triclinic lattices only. Taking into account the reduction of symmetry shows that the symmetry of a given lattice is reduced to the triclinic one when the orien-tation of the magnetic field with respect to the lattice lifts any point-symmetry element except identity, and inversion if the latter is present in the lattice symmetry group. Never-theless, in many experiments the magnetic field is aligned with a high-symmetry direction of the lattice that allows in general keeping some other point-symmetry operations when the magnetic field is applied. For example, the zinc blende lattice has three orthogonal S4 axis. Performing a complete symmetry analysis with taking into account the reduction of the lattice symmetry under the field [8] allows showing that when the field is parallel to one of the S4 axis, the axis keeps its symmetry. The same result also holds for quantum wells (QWs) grown along a S4 direction with the

field being parallel to the direction: the S4 axis perpendicu-lar to the layer plane is kept. In the wurtzite lattice, the c-axis has the C6v symmetry that transforms into the C6 one under a field parallel to the axis. Wurtzite QWs grown along the c-direction have a C3v axis perpendicular to the layer plane that transforms into a C3 axis under a field par-allel to the c-direction. Therefore, the formalism of mag-netic translations does not allow in the general case taking into account the effect of point-symmetry operations [8]. To my 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 QWs [6]. That perhaps arises from the weaknesses of the formal-ism that have been mentioned above as well as from its complexity. In addition, the formalism has been shown in Refs. [1–7] to provide three-periodic electron wavefunc-tions. Indeed, under their restricted conditions, the authors of Refs. [1–7] showed that it is possible to keep the full translational symmetry of the electron Hamiltonian and found such eigenfunctions. Besides, Trellakis [9] consid-ered a singular gauge transformation based on a lattice of magnetic flux lines, Cai et al. [10] proposed a formulation of electronic structure calculation based on a plane-wave basis and Fourrier transform approximation, and Nanciu [11] dealt with Wannier functions. The three authors pro-

We previously established the transformations (including the

gauge transformations) induced within the electron Hamilto-

nian by the point- and space-symmetry operations of three-

and low-periodic lattices under a uniform magnetic field

[P. Tronc and V. P. Smirnov, Phys. Status Solidi B 244, 2010

(2007)]. We also determined the full symmetry group of elec-

tron eigenfunctions by taking into account the electron con-

finement in tubes whose axes are parallel to the magnetic

field direction. The confinement is known from experiments

with semiconductors. From the symmetry point of view, the

confinement implies a lack of translational symmetry in the

directions that are not parallel to the field. A theoretical proof

is given here for the lack of translational symmetry in any lat-

tice and the relation between the electron-Hamiltonian sym-

metry group and the eigenfunction symmetry group is pro-

vided.

Page 2: Exact space symmetry of electron states in lattices under a magnetic field

Phys. Status Solidi B 246, No. 6 (2009) 1249

www.pss-b.com © 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Original

Paper

posed also three-periodic electron eigenfunctions and pro-vided symmetry groups neither for the electron Hamilto-nian nor for the electron wavefunctions. Summing up, it appears that any of the authors of Refs. [1–7, 9–11] proposed three-periodic electron eigenfunctions. Such wavefunctions are clearly unphysical (having no physical meaning) since it is well known from experiments with semiconductors that the actual wavefunctions are confined in tubes whose axes are parallel to the magnetic field direc-tion. Recently, we pointed out the limitations of the works using the formalism of magnetic translations [8]. To elabo-rate a more complete theory, we took into account the transformations (including the gauge transformations) in-duced within the electron Hamiltonian by the point- and space-symmetry operations of three- and low-periodic lat-tices under a magnetic field. Basing on the experimentally well known electron confinement in tubes whose axes are parallel to the magnetic field direction in semiconductors, we determined the exact symmetry group of electron ei-genfunctions. From the point of view of symmetry, the electron confinement in tubes whose axes are parallel to the magnetic field direction implies the lack of transla-tional symmetry in the directions that are not parallel to the field. A theoretical proof is given hereafter of the above lack of translational symmetry for the wavefunctions in any lattice. In addition, the relation between the electron-Hamiltonian and electron-wavefunction symmetry groups is considered. The present paper is organized as follows. Section 2 considers the reduction of wavefunction transla-tional symmetry in three-periodic structures whereas Sec-tion 3 deals with low-peridic ones. Finally, Section 4 pro-vides a brief summary of the results and points out the difference with conventional situations in which the elec-tron-Hamiltonian and electron-wavefunction space groups do coincide. 2 Three-periodic structures The direction of every B vector can be approximated to arbitrary precision as that of a lattice translation that is chosen as the direction of the a3 primitive lattice translation and the x- and y-axis are cho-sen to be parallel to the a1 and a2 lattice primitive transla-tions, respectively. The full symmetry group G* of the electron Hamiltonian involves the geometrical symmetry elements of the group structure G under the field, the ap-propriate gauge transformation being associated to the geometrical element [8]. The group G* is isomorphic to the group G. The group G is the intersection of the symme-try groups of the structure G(c) and of the field F. The latter group involves any translation, the rotations around axes parallel to B, the symmetry or glide planes perpendicular to B, and their products. Let g = (R | a) be a symmetry op-eration of the group G where the rotation R is followed by the translation a. The elements g*∈ G* with a = 0 or a par-allel to B are pure geometrical symmetry elements (g* = g) and form for bulk crystals the three-dimensional one-periodic conventional 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 a1, a2 are primi-tive vectors and n1, n2 are integers (proper lattice transla-tion) or half integers (improper lattice translation) [8]. Since the subgroups GRod and

M*T have no common ele-

ments except the identity and M*T is an invariant subgroup

in G*, the group G* can be represented as the semi-direct product:

M Rod** .G T G= Ÿ (1)

Such a result is obvious for proper lattices translations. Besides, if there is a glide plane perpendicular to the mag-netic field, the improper translation is perpendicular to the field and hence the field direction cannot be the direction of a rotation axis in the G group. As a consequence, the improper translation can be associated with any symmetry operation of the GRod group. Under a symmetry operation of the lattice, the electron eigenenergy is kept. Indeed, 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:

, ,

2

( ) ( ) ,

1ˆ( ) ,

2

j j jH E

e eH V

m c mc

µ µΨ Ψ=

Ê ˆ= + + + ◊Ë ¯

r r

p A r s B (2)

where i�= - —p is the momentum operator, s the spin op-erator, j the eigenvalue index and µ enumerates the eigen-states with the same energy. In Ref. [8] it has been shown that under the transformation g–1, Eq. (2) takes the form:

2

,

,

1ˆ[ ( )] ( )

2 2

exp [ ] ( )2

exp [ ] ( ) .2

j

j j

e eV

m c mc

ei

c

eE i

c

µ

µ

Ψ

Ψ

Ê ˆÊ ˆ+ ¥ - + + ◊Á ˜Ë ¯Ë ¯

Ê ˆ¥ - ¥ ◊Ë ¯

Ê ˆ= - ¥ ◊Ë ¯

p B r a r s B

B a r r

B a r r

(3)

Equations (2) and (3) being identical, their solutions corre-sponding to the same eigenenergy Ej are linear combina-tions one of the others. Therefore eigenfunctions

,

( )j µΨ r of Eq. (2) transform into:

,

exp [ ] ( )2

j

µ

ei C

cµ µ µΨ

�¢

¢

Ê ˆ- ¥ ◊Ë ¯ÂB a r r (4)

and keep the Ej eigenenergy. The C(g) matrices are the wavefunction transformation matrices under a symmetry operation g of the structure under the field. It has been shown [8] that a factor system

2 1 2 2 1( , ) exp [ ]

2

eg g i R

Ê ˆ= - ◊ ¥Ë ¯B a a (5)

Page 3: Exact space symmetry of electron states in lattices under a magnetic field

1250 P. Tronc: Exact space symmetry of electron states in lattices under a magnetic field

© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.pss-b.com

ph

ysic

ap s sstat

us

solid

i b

appears in the multiplication law of matrices C(g) under the two consecutive operations g1 and g2. As a consequence, since in the general case ω(g2, g1) is not equal to 1, the Hamiltonian symmetry group is a ray group and its IRs are projective [1–7]. The eigenfunction

,

( )j µΨ r has no translational symme-try in directions parallel to the (x,y)-plane. Indeed let a be a proper lattice translation in the (x,y)-plane and Ψj,µ′ (r) a complete set of orthogonal eigenfunctions spanning the space of the Ej eigenenergy and including Ψj,µ (r). If

,

( )j µΨ r was periodic in the plane, under the a proper lattice translation in the plane the wavefunction in (4) should be proportional to

,

( )j µΨ r with a constant ratio Cµµ whose modulus should be equal to unity whereas Cµ′µ with µ′ ≠ µ should vanish. One would have the relation:

exp [ ] = .2

µµ

ei C

c�

Ê ˆ- ¥ ◊Ë ¯B a r (6)

Relation (6) cannot be fulfilled except when a and B are parallel. As a consequence, the space symmetry of the wavefunctions is described by the conventional group GRod

that is a subgroup of the group G* of the Hamiltonian (see (1)). The present results are valid for nonmagnetic lattices (insulator, semiconductor, metal). When the magnetic field is not applied, their symmetry is described by conventional space, layer, or rod groups for three-, two-, or one-periodic lattices, respectively. The symmetry of ferro-, antiferro-, or ferrimagnetic lattices is described by more sophisticated groups (Shubnikov or color groups). The effect of an ap-plied magnetic field on such lattices is not considered here. Note that the symmetry group of the Hamiltonian in (2) allows determining the symmetry of the eigenfunctions with taking into account the spin–orbit interaction (SOI). Indeed, introducing into the Hamiltonian a term corre-sponding to the energy associated with the SOI would not modify the Hamiltonian symmetry group hence the wave-function symmetry group since the term should be invari-ant under any symmetry operation of the structure under the field. Therefore, to describe the symmetry of the boson (fermion) states, one has to use single (double)-valued IRs of the wavefunction symmetry group. It should be noticed also that the matrix-element value for an optical transition between two states can be changed when one of the states undergoes a lattice translation within the (x,y)-plane. From experiments, it is widely known that eigenstates are con-fined in tubes whose axes are parallel to the magnetic field, that provides for optical transitions a result that is in agreement with the lack of translational symmetry in the (x,y)-plane for the eigenstates. Last, comparing our results with the results of Brown and other authors [1–7], it can be seen that in our full-symmetry model [8], one can also build unphysical three-periodic wavefunctions. Such wave-functions can be made, for example, of linear combinations of an exact wavefunction (having no translational symme-try in the directions that are not parallel to the field) and of

the wavefunctions deduced from the former by any geo-metrical lattice translation and affected of a Bloch-type phase factor.

3 Low-periodic structures 3.1 Two-periodic structures In the same manner as in three-periodic lattices, it is also possible to choose to ar-bitrary precision one of the two lattice primitive transla-tions to be parallel to the field B when the latter lies within the plane with the translational symmetry. Relation (6) still holds and can be fulfilled only when a and B are parallel. Therefore the wavefunction space symmetry is described by the conventional group GRod (see (1)) that is a subgroup of the G* group of the Hamiltonian. The latter group is a layer ray group with projective IRs. When the field direc-tion is not contained within the plane with the translational symmetry, any translational symmetry is lift for the wave-functions since no lattice primitive translation can be paral-lel to B. The wavefunction symmetry is described by a conventional point group. If B is not perpendicular to the plane, the point group is C1 (no symmetry) since neither a proper (or improper) rotation around the field direction nor a symmetry or glide plane perpendicular to the field can exist. When the field is perpendicular to the plane with the translational symmetry, the point group can eventually in-volve some proper rotations around its direction and/or a symmetry plane perpendicular to it. 3.2 One-periodic structures The factor system in (5) is equal to unity since a2 and R2 a1 are always parallel. As a consequence, the Hamiltonian and wavefunction symmetries are described by conventional groups. It is for both the rod group describing the symmetry of the struc-ture under the field when the latter is parallel to the direc-tion of translational symmetry of the structure. Indeed, re-lation (6) shows that the translational symmetry is kept for the wavefunctions since B and a are parallel. In the other cases, the Hamiltonian symmetry is still described by the rod group describing the symmetry of the structure under the field whereas the wavefunction symmetry is described by a point group that is the intersection of the rod group and of the group F of the field. When the field direction is not perpendicular to the direction of translational symme-try of the structure, the point group is C1 since the two groups cannot have common element except identity. When the field direction is perpendicular, the point group can eventually involve a rotation by π around the field di-rection and/or a symmetry plane perpendicular to the field. 3.3 Zero-periodic structures The Hamiltonian and wavefunction symmetries are described by the same point group that is the intersection of the point group describing the symmetry of the structure and of the group F of the field. 4 Conclusion Taking into account the gauge trans-formations under the symmetry operations of a lattice un-

Page 4: Exact space symmetry of electron states in lattices under a magnetic field

Phys. Status Solidi B 246, No. 6 (2009) 1251

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Paper

der a uniform magnetic field allows keeping the full trans-lational symmetry for the electron Hamiltonian. In three- and two-periodic structures the symmetry group of the Hamiltonian is a ray group with projective IRs whereas it is a conventional rod (point) group in one (zero)-periodic structure. Except for one-periodic structures with the field being parallel to the direction of translational symmetry of the structure and for zero-periodic structures, the group de-scribing the wavefunction symmetry does not coincide with the Hamiltonian group and is only a conventional subgroup of the latter. In one-periodic structures with the field being not parallel to the direction of translational symmetry of the structure, the group describing the wave-function symmetry is a point group. In three- and two-periodic structures, it is a conventional subgroup of the Hamiltonian group with conventional IRs. Indeed the translational symmetry is left for the wavefunctions in di-rections that are not parallel to the field. As a result, the wavefunction group is a rod group whose direction is par-allel to the field for three-periodic structures and also for two-periodic structures with the field being within the plane of translational symmetry. It is a point group for two-periodic structures with the field being out of the plane of translational symmetry.

The results obtained above describe a very particular situation since, generally, for structures not submitted to a magnetic field, the Hamiltonian group and the wavefunc-tion group are identical. This particular situation for struc-tures submitted to the field arises from the way the gauge transforms under the symmetry operations of the structure. That should be taken into account when studying Zeeman splitting for example.

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[7] E. Brown, and L. V. Meisel, Phys. Rev. B 13, 5271 (1976).

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