Example of the Theory of Bloch Waves

Revista Brasileira de Física, Vol. 9, NP 2, 1979
On an
Example of the Theory of Bloch Waves*
EDGAR RIEFLIN**
Departamento de FÍsica, ICEx, Universidade Federal de Minas Gerais, Belo Horizonte,
MG, Brasil
Recebido em 2 de Fevereiro de 1978; manuscrito revisto recebido em 19 de Fevereiro de
1979
The Lamé equation
d2$/dz2
- (h+
2 p ( z ) ) $ = O i s i n t e r p r e t e d as
the schr8dinger equation o f a p a r t i c l e i n a one dimensional p e r i o d i c pot e n t i a l and the corresponding band s t r u c t u r e i s discussed.
I n t e r p r e t a - s e a equação de Lamé
mo equação de sch$dinger
d21/dz2
- (h +
2 p ( z ) 11
= O co-
para uma p a r t i c u l a em um p o t e n c i a l p e r i o d i c o
unidimensional e discute- se a e s t r u t u r a de f a i x a s correspondentes.
1. INTRODUCTION AND CONCLUSION
There i s no doubt the usefulness o f examples
i l l u s t r a t i n g ge-
n e r a l t h e o r i e s even i f the examples a r e constructed i n a somewhat a r t i f i c i a l way. One dimensional models have always been used t o demonstratethe
fundamental p r o p e r t i e s o f s o l i d s f o r example, long b e f o r e h i g h l y
aniso-
t r o p i c m a t e r i a l have been s t u d i e d as r e a l i z a t i o n s o f such models.Themode1 o f Kronig and ~ e n n e ~ fl o3r ~e l e c t r o n s i n a one dimensional conductor
i s an example which i s wellknown,
I t seems t o have been unnoticed f o r a
long time t h a t t h e s o l u t i o n o f t h e Lamé equation, f i r s t given by Hermit e 3 , a l s o provides a b e a u t i f u l i l l u s t r a t i o n o f Block's theory
*
of
elec-
P a r t i a l l y supported by Financiadora de Estudos e P r o j e t o s and Organiz a t i o n of Arnerican States.
**
Present address: Fachbereich Physik, U n i v e r s i t i t Regensburg D-8400 Regensburg, West-Germany.
t r o n s i n a l a t t i c e . Recently i t appeared i,i the c o n t e x t o f the Korteweg-de V r i e s equation f o r nonl inear waves i n weakly dispers i v e systems4. ~ h e
f a r reaching g e n e r a l i z a t i o n o f the formulas which a r e used i n t h i s paper
can be found t h e r e and they can be used t o approximate a n y a r b i t r a r y one
dimensional p o t e n t i a 1 5 .
I t seems u s e f u l t o present the l n t e r p r e t a t i o n o f the Lamé equa-
t i o n as a ~ c h r g d i n ~ eequation
r
here i n more d e t a i l because
the m a t e r i a l
i s r a t h e r dispersed i n mathematícal papers on the aforementioned
topic.
The p r e r e q u i s i t e knowledge on E1 1 i p t i c Functions we s h a l l need6 does n o t
prevent the educational use o f o u r example and the e x p l o r a t i o n of
it
in
t h i s sense i s o u r o b j e c t i v e .
The p l a n o f t h i s paper i s the f o l l o w i n g : S t a r t i n g f r o m t h e Lamê
equation and i t s e x p l i c i t e l y g i v e n s o l u t i o n we show how the Lamé o p e r a t o r
can be i n t e r p r e t e d as schr&inger
o p e r a t o r . The d i s p e r s i o n
relation
is
effective
derived e a s i l y f o r Bloch waves. The d e n s i t y o f s t a t e s and the
masses are given. The band s t r u c t u r e i s h i g h l y degenerate because i t cont a i n s o n l y one gap. I n respect t o t h i s we r e f e r t o some r e s u l t s
n e c t i o n w i t h the Inverse Problem.
i n con-
We then discuss the l i m i t i n g cases o f
n e a r l y f r e e and t i g h t l y bound e l e c t r o n s . Using a r e p r e s e n t a t i o n
tentials,
p o t e n t i a l by the s u p e r p o s i t i o n of l o c a l " a t o m i ~p~o ~
t o t i c behaviour o f these " a t ~ m i c ' p~o t e n t i a l s ,
of
the
of
the
t h e asymp-
wavefunction
of
t h e i r bound s t a t e s and f i n a l l y o f the corresponding Wannier f u n c t i o n s i s
given.
2. THE WMÉ EQUATION INTERPRETED AS
SCHRODINGER EQUATION
Though beeing complex, the ?a& equation g i v e n by
w i t h h a parameter, n an i n t e g e r and p the Weierstrassian E l l i p t i c Funct i o n i s a p p a r e n t l y o f the same form as the schr8dinc~erequation o f a p a r t i c l e i n a one dimensional p o t e n t i a l w r l t t e n i n atomic u n i t s ( E = e = m = l )
To i n t e r p r e t e the Weierstrassian p- Function as a p o t e n t i a l
we have t o
r e s t r i c t t h e i r i n v a r i a n t s t o be r e a l . To exclude the unphysical singul a r i t i e s c i f the o r d e r two from the p o t e n t i a l f u n c t i o n one has
t o take
pure r e a l and pure imaginary p e r i o d s 2wl and 2w3. On the s t r a i g h t l i n e
z = x
+
w ~ , p i s then r e a l and f r e e from s i n g u l a r i t i e s
.
For simpl i-
c i t y ' s sake we w i l l a l s o r e s t r i c t the parameter n o f eq. (1)
to
one.
Chosing f o r the p o t e n t i a l
which has the r e a l p e r i o d 2wl as l a t t i c e constant and w r i t i n g the energy eigenvelue E i n terms o f t h e n o t a t i o n o f (1)
the ~ c h r g t l i n g e requation (2) i s seen t o be e q u i v a l e n t t o the Lamé equat l o n ( I ) r e s t r i c t e d t o the l i n e z = x
+
w3, 2w3 being the imaginary pe-
r i o d o f p.
3. THE DISPERSION RELATION
The s o l u t i o n o f the Lamé equation a r e wellknown 7 . D e f i n i n g
by p(a) = h , the s o l u t i o n s a r e g i v e n by
-
$ ( z ) = e " ~ ( ~ )~ ( z t a )
a(z)
t h i s can bei e a s i l y v e r i f i e d w i t h t h e a i d o f
definitions
the
of
the
Weierstrass i a n 5- and a- Functions and o f t h e add i t i o n theorems. Thewav e f u n c t i o n s $ ( z ) a r e a l r e a d y almost e x p l i c l t l y
waves,
consequently the wavevector
k
in
the
form o f Bloch
can be e x t r a c t e d from t h e m i n the
f o l l o w i n g way. By the q u a s i - p e r i o d i c i t y o f u a s h i f t o f 4 b y o n e l a t t i ce constant 2w
1
results i n
-m 2w
k
, which rewi th n 1 = <(ul). Comparing wi th the phase vector e
same
shift,
we
can
determine
the
sults from the Bloch theorem for the
wavevector:
k=',i=i-
1 p(a) this establishes a parametric representation
Together with & - 2
of the dispersion relation. Using the Inversion of the p-Funct ion,
which leads to the integral formula of p and a similar formula of 5
which are the Weierstrassian Normal Elliptic integrals of the firstand
second kind we can rewrite the dispersion relation as
-
r7
2 p-I(-2E ) 1
rl!
1
o
with the usual invariants g 2 , g 3 an -2Eo arbitrary. A convenient
choice of E, and therefore the integration constants wi 1 1 be given below. i t is now advantageous to factorize the polynomial 4y3 - g2y - g3
in the usual way into 4 ( y - e l ) ( y - e 2 ) (Y - e 3 ) where e l , e 2 , e 3 obey the
relations e l > e 2 > e and e l + e 2 + e 3 = O . With the choice - 2Eo=e 3 thein3
1 ~ - r by
tegration constants above result in - $ Legendre's relation.
'-?
The final form of the dis~ersionrelation iç then given by
A graph of the dispersion relation for a special choice of parameters
is given in Fig.1.
1.0
0.e
0.c
0.4
o.:
Fig.1
-
The d i s p e r s i o n r e l a t i o n i n a s c a l e d p l o t (us =
m
i IJ,,... ) c o r r e s -
ponds t o t h e e q u i v a l e n t r e l a t i o n f o r a f r e e p a r t i c l e t o compare
t o the
case 2w,=1.
4. THE BA,ND STRUCTURE
'With the a i d of the formula (5) t h e band s t r u c t u r e can be d i s cussed. The integrand i s r e a l f o r a11 energies E >
-
e 3 / 2 and so i s t h e
wavevector f o r i n f i n i t e l y wide energy band. On the c o n t r a r y , i n the r e gion
-
e3/2 > E >
-
e 2 / 2 t h e integrand i s imaginary, so t h a t
the wave-
v e c t o r i s becoming complex. T h i s corresponds t o a gap. S p e c i f i c a l l y
which can be obtained by t h e i n t e g r a l formulas o f t h e h a l f p e r i o d s . Because o f t h i s , the wavevector r e t u r n s t o r e a l values f o r the f i n i t e a l -
lowed energy band
- e2/2
> E >
- e1/2.
F i n a l l y i n the region
-
e1/2 > E
t h e r e a r e no r e a l wavevectors. So t h e band s t r u c t u r e i s h i g h 1y degeneratedand has o n l y one gap as can be seen a l s o i n Fig.1.
t h i s i s g i v e n below i n s e c t i o n
An exph l a n a t i o n o f
7.
Because o u r rnodel i s one dimensional i t i s
easy
to
derive
frorn o u r d i s p e r s i o n r e l a t i o n t h e d e n s i t y o f s t a t e s and frorn t h i s t h e e f f e c t i v e masses.
A d i f f e r e n t i a t i o n o f t h e wavevector w i t h r e s p e c t t o t h e e n e r gy p r o v i d e s t h e densi t y o f s t a t e s :
T h i s e x p r e s s i o n shows e x p l i c i t l y t h e s i n g u l a r b e h a v i o u r o f t h e
o f s t a t e s a t t h e band edges
-
e3/2,
-
e2/2,
-
density
e1/2.
I n a s i m i l a r way we have f o r t h e e f f e c t i v e rnass
S p e c i f i c a l l y t h e e f f e c t i v e rnasses a t t h e band edges - e 3 / 2 , - e 2 / 2 , -e1/2
a r e g i v e n by
r e s p e c t i v e l y.
5. ON THE. INVERSE PROBLEM
The lnverse Problem c o n s i s t s i n determining
the
potential
from a g i v e n band s t r u c t u r e . From t h i s p o i n t o f view we c o u l d h a v e s t a r t e d from a formula f o r band s t r u c t u r e s w i t h o n l y a f i n i t e numberof gaps
g i v e n by ~ a c h s t a d t ' and would have g o t our expression o f t h e d i s p e r s i o n
. r e l a t i o n w i t h o u t any o t h e r knowledge than t h a t o f t h e t h r e e
-
e3/2,
-
e2/2,
-
e1/2.
band edges
Hochstadt t o o has g i v e n a d i f f e r e n t i a l equation
g
f o r the p o t e n t i a l i n the case o f band s t r u c t u r e s w i t h o n l y o n e g a p . From
t h i s d i f f e r e n t i a l equation we can r e c o n s t r u c t our p o t e n t i a l
p(x +
w i t h o u t any a d d i t i o n a l knowledge. Only the band s t r u c t u r e s w i t h
one band gap correspond u n i q u e l y t o one p o t e n t i a l as i s known
no
w3 )
or
from the
theory o f t h i s Inverse ~ r o b l e m " .
This a l l o w s a n a i v e study o f the p o t e n t i a l s t a r t i n g from t h e
pararneters o f the band s t r u c t u r e . The t h r e e parameters e l , e*, e j
above a r e n o t independent.
used
I f b denotes the w i d t h o f the f i n i t e allowed
band and R the w i d t h o f t h e gap, then the r a t i o
e2 - e3
---el
-
e3
=
R
= m
(8)
a,+b
gives m , the square o f t h e modulus which occurs i n the Jacobian E l l i p t i c
Functions. From the modulus the r a t i o o f per iods ca
be c a l c u l a t e d :
wi t h the wel 1 known pa rameter q named nome. The l a t t ce constant
i s de-
termined by
(9)
w i t h the
E l l i p t i c I n t e g r a l of the f i r s t k i n d K. These formulas
g i v e j u s t a s o l u t i o n o f t h e Inverse Problem associated w i t h Weierstrassian E l l i p t
o f Jacobian
Functions. F i n a l l y we express the p o t e n t i a l
i n e amplitude sn:
wi.th t h e a i d
6. THE LIMITING CASES
I t i s i n s t r u c t i v e t o v i s u a l i z e the p o t e n t i a l i n t h e l i m i t i n g
cases o f small gap and o f small f i n i t e band.
I n t h e l i m i t o f a smallgap
we have a modulus near zero and t h e p o t e n t i a l i s approximately g i v e n by
( u s i n g formula 127.01 o f reference 11)
II
The Schrodinger equation i n t h i s approximation i s o f t h e Mathieu
type.
And t h e model i s t h a t o f n e a r l y f r e e e l e c t r o n s .
The opposite l i m i t o f small band corresponds
to
a
modulus
near one and t h e approximation i s (us i n g formula 127.02 o f reference 11)
T h i s i s a so c a l l e d ~ U s c h l - ~ e l l ep ro t e n t i a l holel*
which i s c h a r a c t e r i -
zed by a vanishing r e f l e c t i o n c o e f f i c i e n t and o n l y one bound s t a t e . T h i s
model corresponds t o t l i a t o f t i g h t l y bound e l e c t r o n s f o r the f i n i t e b a n d
and n e a r l y f r e e e l e c t r o n s f o r t h e i n f i n i t e band.
I n concl usion, the Lamé o p e r a t o r can i n t e r p o l a t e c o n t inous 1y
t h e rnodels o f n e a r l y fi-ee and s t r o n g l y bound e l e c t r o n s , b u t o n l y i f t h e
r e l a t i o n between the parameter o f t h e band s t r u c t u r e
constant g i v e n above by eq.(8)
and
the
lattice
and ( 9 ) holds. L e t us r e s t r i c t f u r t h e r t o
neutra1 p o t e n t i a l s i . e . p o t e n t i a l s , whose mean second d e r i v a t i v e i n one
l a t t i c e c e l l i s zero. This gives the c o n d i t i o n 1
+ b
= 1/2
as
can
be
e a s i l y v e r i f i e d . By t h l s we have a one paremeter s e t o f p o t e n t i a l s w i t h
a minimal l a t t i c e constant 2wl = n. The gap i s now a unique f u n c t i o n o f
t h e l a t t i c e constant. Approximations o f t h i s f u n c t i o n
and s t r o n g l y bound e l e c t r o n a r e g i v e n by ( u s i n g
112.01 o f r e f e r e n c e 11 .)
f o r nearly
formulas
900 .O00
free
and
7. THE ASYMPTOTIC BEHAVIOUR OF THE WANNIER FUNCTIONS
L e t us r e t u r n t o formula (11). The small band degenerates i n
r otential hole.
the l i m i t : t o the bound s t a t e o f t h e ~ 8 s c h l - ~ e l l e p
the
cause o f t h e v a n i s h i n g r e f l e c t i o n c o e f f i c i e n t o f t h e l a t t e r
t r o n s w i t h energies above the gap a r e t r a n s m i t t e d
unhindered,
Beelec-
so t h a t
t h e r e can be no h i g h e r gaps. T h i s simple understanding o f the b a s i c f e atures o f our band s t r u c t u r e can be extended t o a r b i t r a r y b and L.Using
a wel 1known ser i e s expansion o f p i n terms o f csc 2 ( ~ e.7)
f
i t can be
shown by simple c a l c u l a t i o n s t h a t the p o t e n t i a l p(z+u3) can be represent e d by a s u p e r p o s i t i o n o f "atomic"
p o t e n t i a l s centered a t
the
lattice
p o i n t s , p l u s a constant. These "atomic" p o t e n t i a l s are, always
ref le-
c i o n l e s s ~ O s c h l - ~ e l l epro t e n t i a l s w i t h o n l y one bound s t a t e g i v e n by
The energy eigenvalue o f t h e bound s t a t e i s
-
1/2 (7i./21u31) 2 .
t i c a l l y the p o t e n t i a l f a l l s o f f e x p o n e n t i a l l y w i t h the
T
/ l u 3 / (lief.12).
decay
Asymptoconstant
I t i s i n t e r e s t i n g t o compare t h i s w i t h the asymptotic
behaviour o f t h e Wannier f u n c t i o n o f t h e f i n i t e band.
In
his
study o f
t h e a n a l y t i c p r o p e r t i e s o f Bloch waves and Wannier f u n c t i o n s i n one d i m e n ~ i o n ' Kohn
~
has g i v e n a method t o determine t h e asymptotic behaviour
o f t h e Warinier f u n c t i o n s from t h e knowledge o f t h e d i s p e r s i o n
relation
o n l y . The decay constants a r e g i v e n by the maximal values o f t h e imaginary p a r t o f the complex wavevector i n the gaps. Consequently
t o c a l c u l o t e the wavevector where t h e imaginary d e n s i t y o f
through zero.
one
has
s t a t e s goes
I n o u r case t h i s i s E. = ~ 1 ~ / 2 and
w ~ we have
I f we denote w i t h ~ ( m )and ~ ( m )the complete E l l i p t i c I n t e g r a i s
f i r s t and second k i n d and w i t h
o f the
Z ( B , m ) t h e Jacobian Zeta Function thenwe
can g i v e t h e e v a l u a t i o n o f t h e i n t e g r a l above by t h e
following
simple
expression
T h i s @ maximises t h e Zeta F u n c t i o n a t f i x e d modulus ( f o r m u l a
reference 11)
.
141.25 o f
I f we r e p r e s e n t t h e decay c o n s t a n t o f t h e wavefunct i o n o f
t h e bound s t a t e o f t h e p g s c h l - ~ e l l e r p o t e n t i a l
as
a
function o f
s p e c t r a l parameters i n an analogous manner as t h e l a t t i c e
constant
f o r m u l a (9) we o b t a i n :
- = ?r
4n-m)- 2
Fiil.2
-
Dccay c o n i t i i n t s (Nocatiurr
9
2 K' (m)
2lyl
i 5
cxplained
i n thc
tcxt).
the
in
K1(m) being K(1-m). The f a c t o r /s(i+a) occurs a l s o i n t h e expression o f
corres-
the decay constant o f t h e Wannier f u n c t i o n . T h i s comnon f a c t o r
ponds t o t h e decay constant o f t h e Wannier f u n c t i o n s i n t h e t i g h t - b i n d ing l i m i t i n t h e case where the bound s t a t e o f t h e corresponding
mic" p o t e n t i a l l i e s a t the energy
-
" ato-
e1/2 i e . a t the bottom o f our f i n i -
t e band13. The d i f f e r e n t decay behaviour o f the Wannier f u n c t i o n and t h e
wavefunction o f t h e bound s t a t e i s determined by the d i f f e r e n t f a c t o r s
Zmax and a/2K1, r e s p e c t i v e l y , which a r e shown i n Fig.2.
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1
. R.
de L. Kronig and W . G . Penney, Proc. Royal Soc. London 130, 499
(1931).
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2nd
Ed.
Vol.
2411, 379 (1933).
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118 and Vol. IV,
8 (Gauthier V i l l a r s , P a r i s 1905-1917).
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V.B.Matveev
and S.P.Novikov,
Russian Math. Surveys,
.
31, 59 (1976)
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A
CompZex
VariabZe, Vol . I l I , Chapter 5 ( ~ r e n t i c eH a l l , Englewood C1 iffs,1967).
7. E.T.Whittakerand
G.N.Watson,
ACourseofModernAnaZysis,
Ed. (Cambridge U n i v e r s i t y Press, 1927)
bth
.
8. A. Hochstadt, Math. Z e i t s c h r , 82, 237 (1963).
9. A. Hochstadt, Arch. Rat. Mech. Anal.,
19, 353 (1965).
10. H.P.McKean and P. van Moerbecke, Inventiones Mat.,
11
. P.F. Byrd and M .D .Fr i edman,
30, 217(1975).
Handbook of E Z Z i p t i c I n t e g r a Z s f o r
g i n e e r s and Scdentists, 2nd Ed. (Springer Verlag, New York, 1971)
12.
s.
1974)
~
igge,
l P r a c t i c a Z k t w n Mechanics,
.
13. W. Kohn, Phys. Rev.,
215, 809 (1959).
En-
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(Springer Verlag, NewYork,