Haug H., Koch S. Quantum theory of the optical and electronic properties of semiconductors

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Optical Properties of a Quasi-Equilibrium Electron–Hole Plasma 289

Even though Eq. (15.17) has the same form as the free-carrier suscepti-

bility in Chap. 5, the single-particle energies in Eq. (15.17) are the renor-

malized energies calculated in the quasi-static approximation of Chap. 9.

Hence, Eq. (15.17) includes the band-gap renormalization eﬀects due to

the electron–hole plasma. Next, we introduce a vertex function Γ

which

describes the deviations of the full susceptibility χ from χ

=Γ

. (15.19)

The vertex function obeys the equation

=1+

cv,k





s,k,k



. (15.20)

vertex integral equation

The integral equation (15.20) will now be solved by various methods.

The integral over k



is approximated by a discrete sum





→





2π

→



∆k



2π

. (15.21)

In numerical evaluations, one typically obtains converging solutions if

around 100 terms are included in the summation. Here, the k

can be taken

equidistantly, however, better accuracy is obtained if the k

are taken as

the points of support of a Gaussian quadrature. At low densities the angle-

averaged potential becomes singular for k



= k. This singularity has to be

removed before the numerical matrix inversion is performed. One adds and

subtracts a term







k,k



s,k,k









s,k,k



− F

k,k



, (15.22)

where F

k,k



is chosen so that F

k,k

=1and that the sum in the ﬁrst bracket

can be evaluated analytically, e.g., in 3D

k,k



2

+ k

)

. (15.23)

The bracket of the diﬀerence term in (15.22) vanishes at k = k



where

s,k,k



is singular, so that the total term is a smooth function at k = k



.With

January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2

290 Quantum Theory of the Optical and Electronic Properties of Semiconductors

-2

-4

-2 0 2

Absorption (10 cm )

4-1

3.2

3.4

3.6

3.8

-4 -2 0 2

(h - E )/Ew

Refractive Index

a) b)

Fig. 15.1 Computed absorption (a) and refractive index spectra (b) for bulk GaAs at

T =10K using the matrix-inversion procedure. The used parameters are: m

=0.0665,

=0.457, 

=13.74, 

∞

=10.9, a

=125Å, E

=4.2 meV, and the densities are 0

(1), 5 · 10

−3

(2), 3 · 10

−3

(3), and 8 · 10

−3

(4), respectively. The damping

is γ =0.05E

such a procedure, compensation terms can be found for the plasmon–pole

approximation both in 3D and 2D.

Fig. 15.1 and Fig. 15.2 show the optical spectra which are obtained by

numerical matrix inversion for the examples of bulk GaAs and bulk InSb

for various plasma densities. The resulting complex susceptibility

χ(ω)=



∗

(15.24)

deﬁnes the complex optical dielectric function (ω)=1+4πχ(ω) which in

turn determines the spectra of absorption α(ω), Eq. (1.51) and refraction

n(ω), Eq. (1.50).

At low plasma densities the absorption spectra in Fig. 15.1-a shows well-

resolved 1s and 2s exciton resonances, followed by an ionization continuum

enhanced by excitonic eﬀects. A plasma density of 10

−3

is close

to the Mott density where the exciton bound states cease to exist. The

excitonic enhancement still causes the appearance of a maximum around

the original exciton ground-state position. At still higher densities a band-

gap reduction far below the position of the original exciton ground state is

seen, and simultaneously a build-up of optical gain occurs. The vanishing of

the exciton resonance in the absorption spectrum also causes considerable

changes in the refractive index, as shown in Fig. 15.1-b. Particularly, if

the laser beam is tuned below the exciton resonance, where the absorption

is relatively weak, the index of refraction decreases with increasing plasma

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Optical Properties of a Quasi-Equilibrium Electron–Hole Plasma 291

-10 0 10 20 30

Absorption (10 cm )

3-1

(h - E )/Ew

Fig. 15.2 Computed absorption spectra for bulk InSb at T =77Kusingthematrix-

inversion procedure. The used parameters are: m

=0.0145, m

=0.4, 

=17.05,



∞

=15.7, a

=644.8 Å, E

=0.665 meV, and the densities N are 0 (1), 8 · 10

−3

(2), 2 · 10

−3

(3), and 3 · 10

−3

(4), respectively.

density. This eﬀect is often exploited in optical switching devices which are

based on dispersive optical bistability, see Chap. 16.

The bulk InSb spectra of Fig. 15.2 have been calculated with a damping

γ =2E

=1meV, which is twice as large as the exciton Rydberg. This

case is typical for narrow band-gap semiconductors. Here, the exciton is not

resolved in the unexcited medium and causes only a step-like absorption

edge.

The corresponding absorption spectra for ideal 2D GaAs are shown in

Fig. 15.3. Only one hole band was taken into account. Experimentally

-2

-4 -2 0 2

(h - E )/Ew

Absorption (10 cm )

4-1

Fig. 15.3 Computed absorption spectra for 2D GaAs at T =77Kusingthematrix-

inversion procedure. The used parameters are: a

=62, 5 Å, and E

=16, 8 meV, all the

other parameters are the same as in Fig. 15.1. The densities N are 0 (1), 1 · 10

−2

(2), 5 · 10

−2

(3), and 1 · 10

−2

(4), respectively.

January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2

292 Quantum Theory of the Optical and Electronic Properties of Semiconductors

in quantum-well structures, one often sees superimposed the contributions

of more than one band, e.g. of the heavy- and the light-hole bands in

GaAs-based materials.

Finally, we discuss the plasma density-dependent absorption spectra of

a quantum wire. As already mentioned in Chap. 8, the screening of the

Coulomb potential by the conﬁned plasma is expected to be of little impor-

tance in a quantum wire, because the ﬁeld lines passing through the barrier

material cannot be screened. Fig. 15.4-a shows the calculated absorption

spectra for a cylindrical GaAs quantum-well wire and Fig. 15.4-b shows the

corresponding refractive index spectra. The q1D Coulomb potential of a

cylindrical quantum wire, Eq. (7.78), is used in the calculations. The spec-

tra have been obtained by matrix inversion and have been calculated with-

out screening. The spectra calculated with screening (Benner and Haug,

1991) do not diﬀer substantially, showing that for q1D systems the eﬀect

of state ﬁlling is the most important source of the optical nonlinearities.

Re X( )w

-10

-9

-8

-7

-6

-4

-3

-2

-1

-5

0.6

0.7

0.5

0.4

0.3

0.2

0.1

0.0

-0.1

0.6

0.7

0.5

0.4

0.3

0.2

0.1

0.0

-0.1

-10

-9

-8

-7

-6

-4

-3

-2

-1

-5

Im X( )w

a) b)

(h - E )/Ew

Fig. 15.4 Computed absorption (a) and infractive index spectra (b) for a cylindrical q1D

GaAs quantum wire at T =300K using the matrix-inversion procedure. The parameters

are the same as in Fig. 15.1. The 1D-densities are na

=0(1), na

=0.5 (2), na

(3), and na

=2(4), respectively. Ω is the energy spacing between subbands, and E

and a

are the bulk exciton binding energy and Bohr radius, respectively.

Studying the absorption spectra in Fig. 15.4-a, one might conclude that

the band-gap reduction is not very strong in quantum wires. Actually,

however, the opposite is true. The band-gap shrinkage in q1D is larger

than in higher dimensional materials, as can be seen in Fig. 15.5.

For the density na

=2, ∆E

reaches already ﬁve exciton binding

energies. The fact, that the absorption peak in Fig. 15.4-a shows only a

slight red shift is thus again a result of the compensation between reduction

of the exciton binding energy and gap shift.

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Optical Properties of a Quasi-Equilibrium Electron–Hole Plasma 293

T=300 K

/E =10

cylindrical wire

-10

-20

-30

E/E

Fig. 15.5 Band-gap reduction, ∆E

, and chemical potentials as functions of carrier

density for the quantum wire of Fig. 15.4

We conclude, that with the reduction of the dimension in mesoscopic

semiconductor microstructures the inﬂuence of screening on the quasi-

equilibrium nonlinearities decreases, while the inﬂuence of state ﬁlling in-

creases.

15.2 High-Density Approximations

The integral equation (15.20) can be solved approximately if the attrac-

tive electron–hole potential is not too strong, i.e., in the high-density

limit, where plasma screening and phase-space occupation have reduced

the strength of the Coulomb potential suﬃciently. We rewrite Eq. (15.20)

by introducing a formal interaction parameter σ, which will be assumed to

be small

=1+





s,k,k



. (15.25)

A power expansion of Γ in terms of σ yields



, (15.26)

where the ﬁrst coeﬃcient is determined by





s,k,k



. (15.27)

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294 Quantum Theory of the Optical and Electronic Properties of Semiconductors

In general, one can express Eq. (15.26) as the ratio of two polynomials

(N,M)



n=0



m=0

, (15.28)

which represents the (N, M)-Pad´e approximation. The coeﬃcients r

and

can be obtained by a comparison with the original expansion (15.26).

We use the simplest nontrivial (0,1)-Pad´e approximation

= P

(0,1)

1 − q

, (15.29)

which results in an optical susceptibility of the form

χ(ω)=



∗

1 − q

. (15.30)

Pad´e approximation

The denominator in Eq. (15.30) expresses the inﬂuence of the multiple

electron–hole scattering due to their attractive interaction potential V

which causes the excitonic enhancement.

Another approximation for the integral equation (15.20) is obtained by

inserting a dominant momentum, which we take as the Fermi momentum,

into the angle-averaged screened Coulomb potential. The Fermi momentum

=(3π

1/3

in 3D (15.31)

=(2πn)

1/2

in 2D, (15.32)

see Chaps. 6 and 7. Then the susceptibility integral equation (15.16) be-

comes

= χ







s,k





= χ

[1 + S(ω)] , (15.33)

and for

S(ω)=





s,k



(15.34)

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Optical Properties of a Quasi-Equilibrium Electron–Hole Plasma 295

we get the self-consistency equation

S(ω)=



s,k



1+S(ω)



(15.35)

with the solution

1+S(ω)=

1 −



s,k

. (15.36)

Eq. (15.36) yields the optical susceptibility

χ(ω)=



−



s,k

(15.37)

expressing again, in a diﬀerent approximation, the eﬀect of the excitonic en-

hancement, however, with a simple momentum-independent enhancement

factor 1+S(ω).

-0.4

0.4

0.8

1.2

1.6

-10 -5 0 5 10 15 20

(h - E )/Ew

Absorption (10 cm )

4-1

Fig. 15.6 3D absorption spectra for GaAs at 300 K for the densities 1 · 10

−3

(1),

1· 10

−3

(2), 2·10

−3

(3), and 3·10

−3

(4), respectively. Matrix inversion (full

lines), Pad´e approximation (dashed lines), high-density approximation (dotted lines).

Taking the spectral representation, Eq. (15.18), only for the imaginary

part, we compare absorption spectra for 3D and 2D semiconductors in

Figs. 15.6 and 15.7 for the diﬀerent levels of approximations. We show the

results obtained a) by solving Eq. (15.20) by numerical matrix inversion (full

curves), b) by using the (0,1)-Pad´e approximation (15.30) (dashed curves)

and c) using the high-density approximation (15.37) (dotted curves).

January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2

296 Quantum Theory of the Optical and Electronic Properties of Semiconductors

The agreement of both approximations b)andc) with the numerical solu-

tion a) becomes very good, if a pronounced optical gain exists. At lower

plasma-densities, where the gain vanishes, but where the densities are still

above the Mott density, the (0,1)-Pad´e approximation yields a slightly bet-

ter description of the excitonic enhancement than the high-density approx-

imation, particularly in the 2D case. At low densities, where bound states

of the exciton exist, both approximations are unable to give a reliable de-

scription of the bound-state resonances in the absorption spectra.

-0.4

0.4

0.8

1.2

1.6

-8 -4 0 4 8

(h - E )/Ew

Absorption (10 cm )

4-1

Fig. 15.7 2D absorption and gain spectra for GaAs at 300 K for the densities

1 · 10

−2

(1), 1 · 10

−2

(2), 2 · 10

−2

(3), and 3 · 10

−2

(4), respec-

tively. The full lines are the result of the matrix inversion, the dashed line is the Pad´e

approximation and the dotted line represents the high-density approximation.

15.3 Eﬀective Pair-Equation Approximation

For many practical applications, it is very useful to have an approximate

analytical solution for the full density regime, which can then be used in

further studies, e.g., of optical bistability (see Chap. 16) or nonlinear optical

devices. Such an approximation scheme (plasma theory) for 3D systems has

been developed by Banyai and Koch (1986) in terms of an eﬀective electron–

hole–pair equation. The main approximation of this theory is that it ignores

the reduction of the attractive electron–hole Coulomb interaction via occu-

pation of k-states by the plasma, assuming that the dominant weakening

comes through the plasma screening. This assumption is reasonable in 3D

systems at elevated temperatures, but it fails in 2D and q1D where state

ﬁlling is more important.

January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2

Optical Properties of a Quasi-Equilibrium Electron–Hole Plasma 297

We start from the Fourier-transformed polarization equation in the form



(ω + iγ) −H



P = −(1 − f

e,k

− f

h.k

E , (15.38)

where the eigenvalue equation of the hermitian Hamilton operator is

Hφ

ν,k

=(

e,k

+ 

h,k

)φ

ν,k

−





s,k−k



ν,k



= E

ν,k

. (15.39)

Expanding the polarization components into the eigenfunctions



ν,k

, (15.40)

the solution of Eq. (15.38) can be written in the form

= −



ν,k

tanh



(ω − E

− µ)



(ω + iγ) − E





∗

ν,k



, (15.41)

whereweapproximate

1 − f

e,k

− f

h,k

 tanh



(ω − E

− µ)



The resulting absorption coeﬃcient is then

α(ω)=

4π

tanh



(ω−E

−µ)





|ψ

(r =0)|

δ(E

− ω) .

(15.42)

The remaining problem for the analytic evaluation is to obtain the eigen-

functions ψ

and the corresponding eigenvalues for the screened Coulomb

potential. Unfortunately, we do not know an analytic solution for the Wan-

nier equation with the Yukawa potential. However, there is a reasonably

good approximation to the screened Coulomb potential, which is the so-

called Hulth´en potential

(r)=−

κ/

2κr

− 1

, (15.43)

for which one can solve the Wannier equation (10.35). As in Chap. 10, we

make the ansatz

(r)=f

(r)Y

(θ, φ) , (15.44)

January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2

298 Quantum Theory of the Optical and Electronic Properties of Semiconductors

where ν = n, l, m represents the relevant set of quantum numbers. We

obtain the radial equation

−

∂

∂r

−

∂f

∂r



l(l +1)

−

gλ

λr

− 1



= 

(15.45)

with

λ =2κ, g =

, and 

Since we are only interested in those solutions which do not vanish at the

origin,

(r =0)=0 ,

we restrict the analysis of Eq. (15.45) to l =0and drop the index of f and

. For this case, we obtain from Eq. (15.45)

−

∂

∂r

−

gλ

λr

− 1

= u , (15.46)

where we introduced

u(r)=rf(r) . (15.47)

Deﬁning

z =1− e

−λr

,w(r)=u(r)/z(1 − z)

(15.48)

with

β =



−/λ

(15.49)

and inserting these deﬁnitions into Eq. (15.46) yields

z(1 − z)

∂

∂z

+[2− (2β +3)z]

∂w

∂z

− (2β +1− g)w =0 . (15.50)

This is the hypergeometric diﬀerential equation, which has the convergent

solution

F (a,b,c; z)=1+

a(a +1)b(b +1)

c(c +1)

+ ... , (15.51)

where it is required that c =0, −1, −2,... .Inourcase,wehave