Dielectric Functions of Wurtzite and Zincblende Structure GaN
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E(qTw) = 1+
I mAk,c,v E(
(k)
-
E,(k + q
-
[hw + ia]2
]
(1)
Where c and v represent the indices of empty conduction and occupied valence bands, respectively, and (Ak) 3 is a reciprocal space volume associated with each mesh element in the first Brillouin zone. The sum is over all mesh elements. In this work the band energies Ec(k) and Ev(k) were determined in the framework of the empirical pseudopotential method[4], 601 Mat. Res. Soc. Symp. Proc. Vol. 395 01996 Materials Research Society
and the overlap integrals, (k, clk + q',v), were evaluated using pseudowavefunctions. The pseudopotential form factors for GaN were empirically determined from experimental data for several critical point bandgaps and the conduction band effective masses[5][1]. Following this procedure, the pseudopotential bandstructure calculation for zincblende GaN yielded results in satisfactory agreement with the published experimental data and with first-principles calculations[6]. For wurtzite structure GaN, however, we had difficulty in fitting all experimental parameters simultaneously. The best fit was achieved by choosing pseudopotential form factors that yielded a bandstructure with all critical point bandgaps larger than the experimental values by approximately 1 eV. This discrepancy was then corrected by shifting all conduction bands rigidly down by 1 eV. Fig. 1 (a) displays the bandstructure for zincblende GaN, and Fig. 1(b) shows the corrected bandstructure for wurtzite GaN. In all calculations the spin-orbit interaction was neglected.
4-4 ____________
-____
r
X
W
L
-81 ___
F
K
A
_____
L
M
r
A
H
K
r
Figure 1: (a). Bandstructure of zincblende GaN with a direct gap of 3.3 eV. (b). Bandstructure of wurtzite GaN with a direct gap of 3.4 eV. Both are produced by the empirical pseudopotential method. Numerical evaluation of equation (1) is straightforward for non-vanishing e. We used a expansion of the overlap integral to obtain the long wavelength limit:
. fl #
2 ~2 lim C(q, w)
=
1 + 2eh --
Ak,c,v
1q•.- 12(Ak)3 [E()
-
EE(f)]{[Ec(f) - E.(/)]2 - [hw + ia]2}
(2)
where 4 is the unit vector in the direction of q, and the overlap integral has been replaced by the matrix element 5gc,= (k, c[l/fik, v), where P is the momentum operator. The static dielectric function c(q- was directly calculated from equation (1). However, for hw > Eg the denominator will be zero at certain /c points, and equation (1) becomes singular. Therefore we first calculated the imaginary part of the dielectric function C2 (q', w) and then used the Kramers-Kronig relation to calculate the real part 61 (, w). From equation (1) the expression of c2 (Ww) has the form
(E W)= -iq (q,
2
ikrq~
I(k,cl +,v)2(Ak)36(Ec(k) 602
-
Ek(f•+-
hw)
(3)
To evaluate 6 2 (q, w), we replaced the delta function by a Gaussian function. Using the Kramers-Kronig relation, we obtained immediately the real part of the dielectric function c1(',w). We will discuss the Brillouin zone discretization schemes and the bands included for zincblende structure and w
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