Applied Mathematics, 2013, 4, 1287-1289
http://dx.doi.org/10.4236/am.2013.49173 Published Online September 2013 (http://www.scirp.org/journal/am)
Analytical Study of Band Structure of
Material Using Relativistic Concept
E. I. Ugwu1, M. I. Echi2
1Department of Industrial Physics, Ebonyi State University, Abakaliki, Nigeria
2Department of Physics, University of Agriculture, Makurdi, Nigeria
Received March 21, 2013; revised April 21, 2013; accepted April 30, 2013
Copyright © 2013 E. I. Ugwu, M. I. Echi. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
In this paper, we present the study of band structure relativistically. Here, Dirac equation is formulated from Hamilto-
nian in which the formulation is found to contain a correction term known as spin-orbit coupling given as

22 22
24
V
mc mc

V xP

  


r
tivistic Hamiltonian and reveals the fact that the classifi-
2. Theoretical Frame Work: Band Theory
e
that modifies the non-relativistic expression for the same formulation. This
term leads to double spin-degeneracy within the first Brillioun zone which is a concept that is not found in other method
of study of band structure of material.
Keywords: Bandstructure; Hamiltonian; Dirac Equation; Spin-Orbit Eigenfunction; Relativistic Concept;
Spin-Degeneracy; Wavefunction; Correction-Term
1. Introduction
The study of band structure of materials non-relativistic
concepts had a long time issue from the Bloch theorem
of the form applied to Schrodinger
equation in which the obtained eigenstates and eigenval-
ues are periodic functions in k-space of which the infor-
mation obtained from the reciprocal lattice gave the in-
formation on the band structure of the material [1-4].



i
ekr rU 


In recent time, more work has been veraciously geared
towards experimental study of band structure specially
with regard to binary and ternary compound due to tech-
nological need of these materials with good band gap for
use in solar cell and optoelectronics [4-6]. None of this
approach coupled with other methods that had been on
ground such as KKR, Pseudopontetial [6,7] etc. recog-
nized spine-orbit.
However, it is important to mention here that in order
to get a through picture of band structure of materials
clearly, non-relativistic concept has to come in as it brings
into consideration the spin-orbit splitting that was neglec-
ted in the relativistic study of band structure.
In this work, we present the analytical study of band
structure of material using Dirac equation formulated
from Hamiltonian in which case it will be found that the
spin-orbit coupling term modifies the original non-rela-
cation of energy band by symmetry is completely altered
as the spin-orbit term involving 2 × 2 Pauli’s matrices is
introduced and that it is no longer a question of phase
shift of the wave function alone.
Band structure as presented by KKR and APW and som
methods depicted precisely the form of the secular equa-
tion in which Bloch wave function is expanded directly
in plane wave. These two methods are related in a man-
ner such that



2
det kK k KE 
 


(1)
has non-trivial solution in which the expression for Ham-
iltonian gives


 

2
expexp i
mn mn
kk Vrkk  (2)
with respect to matrix elements written in terms of p
kHk
lane
wave where


exp i
k
VrV kr



(3)
In the case of KKR, emphasis is give
st
n on lattice
ructure and the scattering properties of a single Muffin-
tin potential while APW centered on secular equation
although both can be related when the phase shift analy-
C
opyright © 2013 SciRes. AM

E. I. UGWU, M. I. ECHI
1288
sis in which the scattering potential is of muffintin in
nature.
In this case, a solution requiring incident plane wave,
ob
i
ekr
the c
 together with the scattered out going waves with
ondition that

0rV r  as r is satisfied.
Under this condi functthod is used to tion, Greenion me
tain


exp i
1d
2π
kr rVrr r
rr




 


(4)
for which the required solution is

2
expiexp id
2π
m
 kzk rVrr



 (5)
That can be written in the form
 
i
ie
e
kz
kz
rf
z


  (6)
In the case of relativistic concept, we start with Ham-
iltonian form of Dirac equationwhich is written as
t





 (7)
where

expresses four component wave function


(8)
And
 



1
2
3
4
kr
rkr
kr








kr

2
H
cyPy mcVr

  (9)
P is momentum operator ir


 while is scalar
poten glected

Vr
tential. However, vector potial is ne as it is
considered to be zero [8]. y

is 4 × 4 matrix expressed
simply in the form
10 0


0
0
01 001, 2, 3
0
001 0
00 01
j
jj
yj













(10)
j

being Pauli’s spin matrices



(11)
With this, we write
12
i
,
10i 0



 
 
01 0
 





(12)
where

and

have components
4
13
2
,
x
x
x
x

 
 
 

(13)
Then according to Schiff, we can write
2
iEmc
t









 (14)
We find that


20mcVc P
  E

 (15)
and


0EV P 

  (16)
However, at low-energy limit,

 a situation 
where


~vc

 where v is the velocity of electron
wavefunscribed b ction as dey

. Based on the elimina-
tion of

in favour of

we ve ha

1
1EV
2
i
22
EP PV
mmc
 






 

 (17)
Using the relation
1
22
11
22
EV EV
mc mc






 (18)

iPV VpV

 (19)




iVPVP  

VP

 (20)
This enables usto write Equation (17) as


2
EVP
2
2
22 22
122
44
EVr
mmc
VVP
mc mc



 


 

(21a)
In this solution, we eliminate E from the right hand
si


de and observe that the last two terms are of the order
1
c

and
2
2
e1
137
 
 (21b)
Hence to order 2,

we write 22EV Pm to
obtain


22
22
2
22 22
28
44
PP
EVr
mmc
VV
mc mc
 
P


 
 

(22)
Most interestingly all relativistic corrections as con-
tained in Equation (17) are important when it has to do
with heavy atoms and near the nucleus. In most case the
correction from the second and fourth terms of the equa-
tion are omitted leaving just the simpler Hamiltonian.

E. I. UGWU, M. I. ECHI
1289
 
2
22
22
P
H
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
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