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The Poisson's summation formula is
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q(r) =
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where F(k) is the 2-D Fourier transform of f(r):
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3-D Green1s Function in 3-D Lattices
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We first consider the case where rand R are three-dimensional. A periodic Green's function involves summation of radiation from sources at all the R in the periodic lattice. Let
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__ eikolr-HI L exp(ik . R)-_-------==_ 47flr - RI
where ko = wVJif. is the wavenumber, k is a wavevector, and LEi is a threefold summation over the lattice. In solid state theory, k is a vector in the first Brillouin zone. More generally, we need to evaluate
Gk(r, r') =
ikoIr-r' L exp(ik . R) 47flr - _, -HIRI e_ _ r R
where rand r' are both in the unit cell centered at the origin. The unit cell centered at the origin has R = O. Direct summation of (3.4.40) converges slowly because the decay for large indices is only l/(distance). Without loss of generality, let r > r'. Note that
\72 Gk(r, r')
+ k~Gk(r, r') =
> r',
-<5(r - r')
We use the addition theorem for r
ikohz(kor)Yzm(r)jz(kor')Yz':n(r') z=o m=-Z Then we define VZm (r) such that Gk(r, r') =
In (3.4.43),
e~ I~r-~'I = '" '" 47fr-r ~ ~
1- -'I
z=o m=-Z
r' =
(8', ')
is the direction vector with angular variables (8', '). and Yzm (8', ') are the spherical harmonics. In this section, unlike in other chapters, we shall define
402 3-D Green's Function in 3-D Lattices
them as
Yi (B -+.)
= [(2l + 1) (l- m)!] "2 pm(cosB)eim
+ m)!
where pzm(cosB) is the associated Legendre polynomial as defined in (1.4.37) of Volume 1. The definition in (3.4.45) differs from (1.4.45) of Volume I by a scale factor. The orthonormality relation for the spherical harmonic is
d Jo dB sinBY[~(B, )Yi'm,(B, ) = 611'6mm,
From (3.4.43) and (3.4.46)
VzmCr) = iko
L eikoRhz(kolr R
where r - R refers to the unit vector point from R to r and is in the direction of the spherical coordinate angles, e and <P. Note that G"k(r,O) is a special case of (3.4.47) with
G"k(r,O) =
From (3.4.47), we can interpret Vzm(r) as multipole radiation from the lattice points. Next, use the integral identity
1 r:;; ho(kor) = -=-k 2
where C is a contour (Fig. 3.4.1) that ensures the convergence of the integral. For C, as indicated in (3.4.49), the arg 8 of C obeys the condition,
-- < arg 8 < (3 4 -
where (3 = arg ko. 1 By using (3.4.49), hi1)(kor) = -h6 )' (kor), the recurrence relation
hl+1(kor) =
(2l + 1) k hz(kor) - hz-1(kor) or
and mathematical induction, it can be shown that
hz(kor) =
2 zr Z r:;; ~ 2 ~ko V 1f
d88 exp [ -r
2 2 k0 8 +-2] 48
Next, we split the integral of s into two parts from 0 to E and from E to 00, where E is known as the splitting parameter. Thus
Dlm(r) = D~(r)
'1""'1(1) (-) vim r -
+ Di~(r)
~" eikoIi 1- - RI1.1yj1m (--::-R) ~ k1 ~ r 2 r (:;;
'1""'1(2)(_) vim r
=~" eikoIiI-_Rllolv (--::-R)~ k1 ~ r 2 .lIm r (:;;
k o ds s21 exp [ _ 2s 2 + 4:2 ] -Ir - RI
The quantity D~ (r) of (3.4.54b) can be calculated by direct numerical integration since an exponential decay is endowed in the integral, particularly for large R. For l = 0, it can be evaluated exactly as will be shown in Section 4.3. For the calculation of D~ (r), we let
k ds s21 exp [ - 2s 2 + ~ ] -Ir - RI 4s 2
f(r - R) = i1Yim(r - R)lr - RI 1exp( -Ir - R1 2s 2) L eikoIi f(r - R) = u(r) Ii
(3.4.55a) (3.4.55b)
1 (1) _ Dim (r) _ kt Vii - 2 2
k~) ds s 21 exp ( 4s 2 u(r)
Using Poisson's summation formula (3.4.30) from Section 4.1 and (3.4.55b),
u(r) =
~ L ei(k+K)or F(k + K)
where F(k) is the Fourier transform of f(r). From (3.4.55a) and the property of Fourier transform
F(k) =
dr i1Yim(f')r 1exp( _r 2s2)e-
We use the spherical wave expansion of
e- ikor = 4~ L(-i)ljl(kr)Yi::n(f)Yim(k) 1m
4.2 3-D Green's Function in 3-D Lattices
Then using the orthogonality relation of (3.4.46), we obtain
~ = 47f - Yim(k)
(28 2 ) +2"
--2 )
Putting (3.4.60) and (3.4.57) in (3.4.56), we have
V ( 1) (r) = -1l - 2 1m koY" IJr
d8 exp
2 ( ~) k
48 2
1 .- - _ """' e~(k+K).r
n~ K
7f -~- k+K k+K E (k+K) 1228-II exp [1- 48-1 ~ 3 2Yim V"2
The d8 integration in (3.4.61) can be performed. We get
V(1)(r) = 47f """'Yi (k--;K)ei(k+K).r Ik+KI 2 2 1m kin ~ 1m o K k + K _ ko
1- -1
exp[k~ -lk+KI
(3.4.62) This converges rapidly in K because of the exponential decay in K. To summarize, using (3.4.62), (3.4.53) and (3.4.54b), we have
Vim (r) = vi:,{ (r)
+ V~ (r)