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Spintronics in Nanostructures
Denis Bulaev
Summer Semester 2007
Lectures Exercises

(16877-01)

days time place every Wednesday 10:15-12:00 Ro om 4.1 (Institute of Physics) every Tuesday 10:00-11:00 Ro om 4.1 (Institute of Physics)

Assistant Mircea Trif (office 4.4, e-mail: Mircea.Trif@unibas.ch)

I. SYMMETRY IN SOLID STATE PHYSICS Lecture 1 1. Introduction to Symmetry in Physics

Symmetry is most basic and important concept in physics. Every process in physics is governed by selection rules that are the consequence of symmetry requirements. For example, momentum conservation is a consequence of translational symmetry of space (r r + a, a R3 ), energy conservation is due to translation symmetry of time (t t + , R), and angular momentum conservation is the consequence of spacial rotation invariance. These are so called continuous symmetries. There are discrete symmetries: time reversal symmetry (t -t), spacial inversion symmetry (r -r), charge reversal symmetry (q -q ), etc. In quantum mechanics eigenstate properties of a Hamiltonian (or an other operator which describe a quantum system) and the degeneracy of eigenvalues are governed by symmetry considerations. Beauty of theory group is that we can translate the heavy and complex language of symmetry operations into a language of a very simple linear algebra.

2. Basics of Group Theory

Definition 1. A group G is a (finite or infinite) set of elements g1 , g2 , g3 , . . . having the fol lowing properties: 1) The product of any two elements of the group is itself an element of the group: g1 , g2 G g3 G : g1 g2 = g3 . 2) The associative law of multiplication holds: g1 (g2 g3 ) = (g1 g2 )g3 . 3) Among the elements of a group there is one and only one element, cal led the identity of unit element e, which has the property g e = eg = g g G. 4) Each group element g has an inverse element g -1 such that g g -1 = e: g G, g -1 G : g g -1 = e. In general, the elements of a group do not commute: g1 g2 = g2 g1 . Definition 2. Group G
A

in which g1 g2 = g2 g1 g1 , g2 G

A

cal led as an Abelian group.

Definition 3. Order of a group G, denoted as |G|, is the number of elements in the set G. Definition 4. A subgroup is a col lection of elements within a group that by themself form a group. Theorem 1. If in a finite group an element x is multiplicated by itself enough times (n), the identity xn = e is eventual ly recovered. Definition 5. The order of an element x is the smal lest value of n such that xn = e. Definition 6. Each element of a finite group may be represented as a power or product of powers of a certain finite number of elements cal led generators of the group. Definition 7. An element b G conjugate to a G is by definition b = xax-1 x G. Definition 8. A class is the total set of conjugated elements. Theorem 2. Al l elements of the same class have the same order. Definition 9. A group consisting of elements e, a, a2 , a3 , . . . , an
-1

(an = e) is said to be cyclic group.


2
3. Point Groups

Any transformation which brings a body into coincidence with itself can be decomposed into elementary transformations: 1) rotation about an axis, 2) reflection in a plane, 3) translation by some vector. The symmetry group of a body of finite dimensions cannot contain a translation, since for any finite body there must be a fixed point, namely, the center mass. Definition 10. A symmetry group in which there is a fixed point common to al l transformations of the group is cal led a point group.

Here is the list of all "elementary" elements of point groups: e is the identity, cn is the rotation through 2 /n angle, is the reflection in a plane, h is the reflection in a horizontal plane (perpendicular to the axis of highest rotational symmetry), v is the reflection in a vertical plane (parallel to that axis), d is the diagonal reflection in a plane through the origin and the axis with the `highest' symmetry, but also bisecting the a i is the inversion (x -x, y -y , z -z ), sn is the improper rotation through 2 /n angle (sn = h cn is the rotation followed by reflection in horizontal plane), icn is the compound rotation-inversion. There are only fourteen typ es of finite point groups: Cn , S2n , Cnh , Cnv , Dn , Dnh , Dnd , T , Td , Th , O, Oh , Y , Yh .