E. I. UGWU, M. I. ECHI

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1289

2

22

22

P

VrV P

mmc

(23)

In this case it is found the only spin-orb

term modifies the original non-relativistic Ham

Th

it coupling

iltonian.

is brings to the focus the idea of the fact that the clas-

sification of energy levels by symmetry is completely

altered as the spin-orbit term involves 2 × 2 Pauli’s ma-

trices. Base on this case it is taken into account especially

in forming the irreducible matrix representations that

influence the symmetry group of the crystal.

From the study on the effect of relativistic corrections

on energy levels in PbTe by Pratt and Ferreira, there was

no indicated evidence on modification of valence and

conduction band extreme [9]. The only observed fact

from their work which was experimental as compared to

what is analytically observed isthat in the absence of

spin-orbit coupling there is double spin-degeneracy at

any point k in the first Brillioun zone with the eigenfunc-

tion generalized as

O

ur

ii

e, e

kv kr

k

k

ur

O

(24)

This degeneracy persists in the presenc

coupling in as much as there exists crystal inversion cen-

tre

e of spin-orbit

in which case it is considered that

EkE k

.

This explains the fact that the degeneracy is not altered

by the presence of spin-orbit in accordance with kp

method when only first order perturbation theory is ap-

plied.

Since the second term on the right hand-side of Equa-

tion (21) denotes relativistic mass velocity correction, the

fourth term has no classical analogue and it is then re-

ferred as the ep term in which case e is referred to as

effective field experienced by the electron. Though the

two last term quation (21) do not introduce any fur-

ther change in the scheme states are classified according

to symmetry, they can cause important corrections to the

band structure observed by Johnson, Conklin and Pratt in

their study of e

PbT [10] from the foregoing analysis,

Dirac equation for electrons in crystals is only solvable in

approximate form of Equation (23) which is a reduced

form of Equation (21).

It should be worth m

in E

entioning here that many of those

m

REFERENCES

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ence of lattice structure and potential on the wave func-

tion as in Equations (3) and (5). Some of those methods

such as APW, KKR and Pseudopotential also looked at

the phase shift in function and quantum defect [11].

ft and N. D. Mermin, “S

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the Theory of Met-

resentations of Space

of Semiconduc-

G. N. Pratt, Physical Re-

cal Solid State

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[8] J. Zak, “The Irreducible Rep

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