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Excitations in Organic Solids$

Vladimir Agranovich

Print publication date: 2008

Print ISBN-13: 9780199234417

Published to Oxford Scholarship Online: January 2009

DOI: 10.1093/acprof:oso/9780199234417.001.0001

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(p.439) Appendix B CALCULATIONS OF POLARIZATION IN INORGANIC QUANTUM WELLS AND IN ORGANICS

(p.439) Appendix B CALCULATIONS OF POLARIZATION IN INORGANIC QUANTUM WELLS AND IN ORGANICS

Source:
Excitations in Organic Solids
Publisher:
Oxford University Press

B.1 General relations

Consider67 two spatially separated systems of charges a and b with charge densities ρa(r) and ρb(r), respectively. Then the interaction energy is given by

(B.1)
V = ρ a ( r ) G ( r , r ) ρ b ( r ) d 3 r d 3 r ,
where G(r, r′) is the Green's function of the Poisson equation. In vacuum it is the usual 1/|rr′|, in an inhomogeneous anisotropic medium with the dielectric tensor ∊ij(r) it satisfies the equation
(B.2)
x i i j ( r ) x j G ( r , r ) ρ b ( r ) = 4 π δ ( r r ) .
If the total charges of the systems a and b are zero, one may introduce the polarizations, vanishing outside the regions of space, occupied by the systems (1), such that
(B.3)
ρ a ( r ) = div P a ( r ) , ρ b ( r ) = div P b ( r ) .
Then the energy may be rewritten as
(B.4)
V = P i a ( r ) P j b ( r ) 2 G ( r , r ) x i x j d 3 r d 3 r = P a ( r ) ( r ) d 3 r ,
where E(r) is the electric field produced by the system b:
(B.5)
i b ( r ) = χ ¯ ( r , r ) P j B ( r ) d 3 r , χ ¯ ( r , r ) 2 G ( r , r ) x i x j .
The Coulomb response function χ ¯ ( r , r ) gives the ith component of the electric field at the point r, produced by the jth component of a point dipole, situated at the point r′; χ ¯ ( r , r ) is nothing else but the limit of χij(r, r′, ω) at ω → 0 (p.440) (electrostatics) or, equivalently, c → ∞ (no retardation). Working in this limit is justified if the characteristic distances in the problem are much shorter than the resonant light wavelength, which is equivalent to k ≫ ω/c.

One should keep in mind that speaking of two different electron densities for the systems a and b already completely disregards the exchange effects, which is justified if the wavefunction overlap between the systems a and b is negligible.

B.2 Polarization in a semiconductor

For the problems under consideration the most interesting are the zinc-blende semiconductors formed by elements of the groups III and V or groups II and VI of the periodic table (like GaAs or ZnSe). In these materials the lowest conduction band is doubly degenerate, and the highest valence band is four-fold degenerate at k = 0 due to the symmetry of the crystal, both bands having extremes at k = 0 (the Γ point). At wavevector k ≠ 0 the four-dimensional manifold splits into two bands (each of them being doubly degenerate), having different effective masses. They are called the light hole and heavy hole bands (2)–(4). For our purposes it will be sufficient to consider the first one a nondegenerate conduction band and one a nondegenerate valence band, assuming the contribution of different bands at the end, when needed.

A system of many identical particles is most conveniently described by the ψ ^ field operators (5). The generic form of the electronic charge density operator in a semiconductor is

(B.6)
ρ ^ sem ( r ) = e ψ ^ ( r ) ψ ^ ( r ) .
Taking into account only two nondegenerate bands, one can split the electronic ψ-operator into two parts, corresponding to the conduction and valence bands:
(B.7)
ψ ^ ( r ) = ψ ^ c ( r ) + ψ ^ v ( r ) ψ ^ e ( r ) + ψ ^ v ( r ) ,
where we introduced a hole creation operator as a valence electron destruction operator. In the operator of the electron density
(B.8)
ψ ^ ( r ) ψ ^ ( r ) = ψ ^ v ( r ) ψ ^ c ( r ) + ψ ^ c ( r ) ψ ^ v ( r ) + ψ ^ c ( r ) ψ ^ c ( r ) + ψ ^ v ( r ) ψ ^ v ( r ) ,
the first two terms correspond to the interband transitions, the last two, to the intraband transitions.

Consider the first term (the second one being just its hermitian conjugate). The matrix element of the operator ψ ^ v ( r ) ψ ^ c ( r ) between the ground state |0> and some eigenstate |s> of a single electron–hole pair is

(B.9)
< 0 | ψ ^ v ( r ) ψ ^ c ( r ) | s > = v 0 d 3 k ( 2 π ) 3 d 3 k ( 2 π ) 3 Ψ s ( k , k ) u c , k ( r ) u v , k * ( r ) e i k r + i k r v 0 u c ( r ) u v * ( r ) Ψ s ( r , r ) .
(p.441) Here v 0 is the unit cell volume, u c(r) and u v(r) are the conduction and valence band extremes Bloch functions,68 and Ψs(r e, r h) is the envelope wavefunction of the electron–hole pair state |s> normalized to the unit integral
| Ψ s ( r e , r h ) | 2 d 3 r e d 3 r h = 1.

Now consider the interband matrix element of the charge density, corresponding to the first term in the expression (B.8). One is interested in the long-wavelength Fourier components of the density, given by

(B.10)
ρ 0 s ( q ) < 0 | ρ ^ sem ( q ) | s > = e d 3 k ( 2 π ) 3 Ψ s ( k , q k ) u . c . u c , k ( r ) u v , k q * ( r ) d 3 r ,
which is obtained from the first line of (B.9) using the periodicity of the Bloch functions. More specifically, due to the periodicity, the product u c , k ( r ) u v , k q * ( r ) contains plane waves with wavevectors equal to either zero or a reciprocal lattice vector. Being interested only in the long-wavelength part of ρ 0s, one should pick up only the zero wavevector contribution, which corresponds to integration over the unit cell, as done in (B.10).

If one simply approximates the Bloch functions in (B.10) by those for the band extremes, the result will be zero due to the orthogonality

u . c . u c ( r ) u v * ( r ) d 3 r = 0.
Hence one can expand the Bloch functions using the k·p perturbation theory (2)–(4), to find the admixture to the functions u c,k(r) of the Bloch functions u b(r) of all other bands b:
(B.11)
u c , k ( r ) u c ( r ) i k j 2 m 0 b < b | / x j | c > E c ( 0 ) E b ( 0 ) u b ( r ) = u c ( r ) i k j b < b | x j | c > u b ( r ) ,
where the symbol <b|O|c> denotes
u . c . u b * ( r ) O u c ( r ) d 3 r .
(p.442) In the transition from the first to the second line of (B.11) the quantum-mechanical relation r ˙ = i ( / m 0 ) for the bare electron in the crystal was used, r ˙ being related to the commutator of r with the crystal Hamiltonian. Expanding u v , k q * ( r ) analogously and substituting them into (B.10), one obtains
(B.12)
ρ 0 s ( q ) = i q d v c d 3 k ( 2 π ) 3 Ψ s ( k , q k ) , ρ 0 s ( r ) = div ( d v c Ψ s ( r , r ) ) ,
where d vc = <v|(−e r)|c> is the transition dipole moment of the unit cell. The contributions of all the bands different from the conduction and valence bands vanish due to the orthogonality. According to (B.3), the obtained charge density corresponds to the interband polarization
(B.13)
< 0 | P ^ ( r ) | s > = d v c Ψ s ( r , r ) .
The expression (B.13) is the basic one to be used below. The Cartesian components of the dipole moments d i v c ( i = x , y , z ) may be expressed in terms of Kane's energy E 0 (3) as
(B.14)
| d i v c | 2 = e 2 2 E 0 c i 2 2 m 0 E g 2 = c i 2 ( e a 0 ) 2 E 0 Ry 0 E g 2 c i 2 | d v c | 2 ,
where a 0 and Ry0 are the hydrogen atom Bohr radius and Rydberg constant, and c i are the appropriate symmetry coefficients. In semiconductors with the zinc-blende structure c x h h = c y h h = 1 , c z h h = 0 (heavy holes), and c x l h = c y l h = 1 / 3 , c l h = 4 / 3 (light holes).

For an intraband transition between the states |s> and |s′> of the electron–hole pair one may simple average ψ ^ c ( r ) ψ ^ c ( r ) and ψ ^ v ( r ) ψ ^ v ( r ) over the unit cell using the Bloch functions at the band extremes, since the principal term does not already vanish. As a result, the corresponding matrix element of the charge density is given by the sum of the electron and hole contributions:

(B.15)
< s | ρ ^ sem ( r ) | s > = e Ψ s * ( r , r h ) Ψ s ( r , r h ) d 3 r h + e Ψ s * ( r e , r ) Ψ s ( r e , r ) d 3 r e .

B.3 Polarization in organics

We assume that the excitations in the organic medium are localized, corresponding to the excited states of a molecule or a group of strongly coupled molecules. Thus, the organic subsytem may be described by the ground state |g A>, and the excited states |r, ν>, where r is the continuous position of the excited state and ν is a continuous quantum number, labeling the excited states at the point r. As we restrict ourselves to the linear regime, only “one-particle” excited states are (p.443) considered, which means that two excitations |r, ν> and |r′, ν′> are not allowed to exist simultaneously. The particular dissipation mechanism, determining the structure of these states need not to be specified here. We use the following normalization of the states

(B.16)
< g A | g A > = 1 ,
(B.17)
< g A | r , ν > = 0 ,
(B.18)
< r , ν | r , ν > = δ ( ν ν ) δ ( r r ) ,
(B.19)
I ^ A = | g a > < g A | + d 3 r d ν | r , ν > < r , v | ,
where ÎA is the unit operator for the organic subsystem. The Hamiltonian and the polarization of the organic medium are written as
(B.20)
H ^ A = | g A > 0 < g A | + d 3 r d ν | r , ν > E ν ( r ) < r , v | ,
(B.21)
P ^ A ( r ) = d ν | r , ν > d ν ( r ) x < r , ν | + h .c . ,
where E ν(r) is the energy of the corresponding state and d ν(r) is the matrix element of the dipole moment between the excited and the ground state:
(B.22)
d ν ( r ) = < r , ν | ( i e i r ^ i ) | g A > ,
with e i, r ^ i being the charge and the position operator of the ith charge in the medium and the sum is taken over all charges constituting the medium. Both E ν(r) and d ν(r) are assumed to be slowly varying in space.

Notes:

(67) See also Agranovich, V. M., Basko, D. M., La Rocca, G. C, and Bassani, F. (1998). J. Phys.: Condens. Matter 10, 9369, and Basko, D. M. (2003). Electronic energy transfer in a planar microcavity. In: Thin Films and Nanostructures. Electronic Excitations in Organic Based Nanostructures 31, edited by V. M. Agranovich and G. F. Bassani. Elsevier, Amsterdam, pp. 403–446.

(68) We adopt the term “Bloch functions” for the cell-periodic part of the full electron wave-function in the crystal.