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 299

a =1+β +



g + β

b =1+β −



g + β

c =2 . (15.52)

15.3.1 Bound states

For the bound-state solutions, we have 

< 0.Sincef must be a normal-

izable function, we request

f(r →∞) → 0 , (15.53)

which yields the condition that

b =1− n, n=1, 2, 3,...

and therefore,

β =

(g − n

) ≡ β

and 

= −λ

; n =1, 2,... . (15.54)

The energetically lowest bound state (1s-exciton) is ionized for

g → 1 , i.e., a

κ → 1 ,

which is the Mott criterion for the Hulth´en potential, Eq. (15.43). Inverting

the transformations (15.47), (15.48), and using Eq. (15.51), we obtain the

bound-state wave functions as

(r)=N

z(1 − z)

F (1 − n, 1+g/n, 2; z)Y

, (15.55)

where the normalization constant is



8π



−



. (15.56)

The relevant factor for the optical response is therefore

|ψ

(r =0)|

32π



−



, (15.57)

where a factor 1/4π comes from the spherical harmonics.

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300 Quantum Theory of the Optical and Electronic Properties of Semiconductors

15.3.2 Continuum states

For the unbound solutions of Eq. (15.45), we have 

> 0 and

= i

√



/λ . (15.58)

Inserting (15.58) into Eqs. (15.51) and (15.52) and normalizing the resulting

wave function in a sphere of radius R, i.e.,

4π



dr|u

(r)|

=1 , (15.59)

we obtain 

= ν

and

= N

(1−z)

1+β



g −|β

, 1+β

−



g −|β

, 2; z

(15.60)

with

√





|β

|sinh(2π|β

|)g

cosh(2π|β

|) − cos

2π



g −|β





1/2

, (15.61)

where cos(ix)=cosh(x). Finally, for the optical response, we need



|ψ

(r =0)|

f(β

2π

(15.62)



∞

sinh(πg

√

cosh(πg

√

x) − cos(π



4g − xg

)

f(x) ,

where the spin summation together with the spherical harmonics contribute

afactor1/2π.

15.3.3 Optical spectra

Combining Eqs. (15.57) and (15.62) with Eq. (15.42) yields

α(ω)=α

tanh



(ω−E

− µ)



ω







−





∆ −





∞

sinh(πg

√

cosh(πg

√

x) − cos(π



4g − xg

)

(∆ − x)



, (15.63)

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Optical Properties of a Quasi-Equilibrium Electron–Hole Plasma 301

where α

and ∆ are deﬁned in Eqs. (5.81) and (10.106), respectively, the

n-summation runs over all bound states,

(x)=

πΓcosh(x/Γ)

, (15.64)

and a factor 2 in the exciton part results from the spin summation.

Eq. (15.63) yields semiconductor absorption spectra that vary with carrier

density N =Σf(k). The theoretical results agree very well with experimen-

tal observations for many diﬀerent semiconductor materials. An example

of such a comparison with experiments is shown in Fig. 15.8. From the

Absorption (x10 /cm)a

1.5

1.2

0.9

0.6

0.3

0.03

0.02

-0.03

-0.02

-0.06

-0.04

-0.06

1.38

1.44

1.50

-10

Energy (eV)

Kramers-Kronig

(h - E )/Ew

Fig. 15.8 Experimental and theoretical absorption and refractive index spectra are com-

pared for room temperature GaAs. [After Lee et al. (1986).] (a) The experimental ab-

sorption spectra are obtained for diﬀerent excitation intensities I(mW):1)0;2)0.2;3)

0.5; 4) 1.3; 5) 3.2; 6) 8; 7) 20; 8) 50 using quasi-cw excitation directly into the band and

a15µm excitation spot size. The oscillations in curve 8) are a consequence of imperfect

antireﬂection coating. (b) The dispersive changes ∆n are obtained through a Kramers–

Kronig transformation of the absorptive changes α(I =0)− α(I) (a). The agreement

with direct measurements of dispersive changes has been tested for the same conditions

using a 299-Å multiple-quantum-well sample. (c) Calculated absorption spectra using

the plasma theory for the densities N (cm

−3

): 1) 10

(linear spectrum); 2) 8 · 10

;3)

2·10

;4)5·10

;5)8·10

;6)1·10

;7)1.5·10

.(d) Kramers–Kronig transformation

of the calculated absorption spectra.

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302 Quantum Theory of the Optical and Electronic Properties of Semiconductors

absorptive changes

∆α(ω)=α(ω,N

) − α(ω, N

) , (15.65)

one obtains the corresponding dispersive changes

∆n(ω)=n(ω, N

) − n(ω,N

) (15.66)

through the Kramers–Kronig transformation

∆n(ω)=



∞

dω



∆α(ω



)

2

− ω

, (15.67)

where P again indicates the principle value. Examples of the dispersive

changes in bulk GaAs are shown in Figs. 15.8-b and 15.8-d.

It is worthwhile to note at this point that the Kramers–Kronig transfor-

mation (15.67) is valid even though we are dealing with a nonlinear system,

as long as ∆α depends only on parameters that are temporally constant.

We have to be sure that the carrier density N does not vary in time, or that

N varies suﬃciently slowly so that it is justiﬁed to treat its time-dependence

adiabatically.

REFERENCES

For details on the matrix-inversion technique see:

H. Haug and S. Schmitt-Rink, Progr. Quantum Electron. 9, 3 (1984) and

the given references

S. Schmitt-Rink, C. Ell, and H. Haug, Phys. Rev. B33, 1183 (1986)

The Pad´e and high-density approximations are discussed in:

C. Ell, R. Blank, S. Benner, and H. Haug, Journ. Opt. Soc. Am. B6, 2006

(1989)

H. Haug and S. W. Koch, Phys. Rev. A39, 1887 (1989)

For the quantum-wire calculations see:

S. Benner and H. Haug, Europhys. Lett. 16, 579 (1991)

The eﬀective pair-equation approximation (plasma theory) is derived and

applied in:

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

Optical Properties of a Quasi-Equilibrium Electron–Hole Plasma 303

L. B´anyai and S.W. Koch, Z. Physik B63, 283 (1986)

S.W. Koch, N. Peyghambarian, and H. M. Gibbs, Appl. Phys. (Reviews)

63, R1 (1988)

Y.H. Lee, A. Chavez-Pirson, S.W. Koch, H.M. Gibbs, S.H. Park, J.

Morhange, N. Peyghambarian, L. B´anyai,A.C.Gossard,andW.Wieg-

mann, Phys. Rev. Lett. 57, 2446 (1986)

PROBLEMS

Problem 15.1: Show that the vanishing of the inversion occurs at the

photon energy ω − E



= µ

+ µ

Problem 15.2: Derive the expression for the optical susceptibility using

the (1,2) Pad´e approximation.

Problem 15.3: Derive the self-consistency equation (15.35).

Problem 15.4: Discuss the Hulth´en potential for small and large radii

and compare it to the Yukawa potential.

Problem 15.5: Verify that Eq. (15.63) yields the correct free-particle

result for a

→∞and the Wannier result for κ → 0.

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Chapter 16

Optical Bistability

Optical instabilities in semiconductors can occur if one combines the strong

material nonlinearities with additional feedback. The simplest example of

such an instability is optical bistability, in which one has situations with

two (meta-) stable values for the light intensity transmitted through a non-

linear material for one value of the input intensity I

. Which transmitted

intensity the output settles down to, depends on the excitation history. A

diﬀerent state is reached, if one either decreases the incident intensity I

from a suﬃciently high original level, or if one increases I

from zero. The

possibility to switch a bistable optical device between its two states allows

the use of such a device as binary optical memory.

A proper analysis of the optical instabilities in semiconductors requires

a combination of the microscopic theory for the material nonlinearities with

Maxwell’s equations for the light ﬁeld, including the appropriate boundary

conditions. The polarization relaxes in very short times, determined by

the carrier–carrier and carrier–phonon scattering, to its quasi-equilibrium

value which is governed by the momentary values of the ﬁeld and the carrier

density. Therefore, we can use the quasi-equilibrium results of Chap. 15 for

the optical susceptibility as the material equation, which implies that the

polarization dynamics has been eliminated adiabatically.

The process of carrier generation through light absorption couples the

electron–hole–pair density to the electromagnetic ﬁeld. The electromag-

netic ﬁeld in turn is described by the macroscopic Maxwell equations, in

which the polarization ﬁeld depends on the value of the electron–hole–pair

density through the equation for the susceptibility. This set of equations,

i.e., the microscopic equation for the susceptibility together with the macro-

scopic equations for the carrier density and for the light ﬁeld, constitutes the

combined microscopic and macroscopic approach to consistently describe

305

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306 Quantum Theory of the Optical and Electronic Properties of Semiconductors

semiconductor nonlinearities.

16.1 The Light Field Equation

Before we discuss two examples of optical bistability in semiconductors, we

present a systematic derivation of the relevant equations. Wave propagation

in dielectric media is described by



∇

− grad div −

∂

∂t



E =

4π

∂

∂t

, (16.1)

where Maxwell’s equations request div D = 0, since we assume no external

charges. However, div E = −E

∇



= −E · ∇ln(),where is the medium

dielectric constant, is generally not zero. Therefore, the electromagnetic

ﬁeld is no longer purely transverse but also has a longitudinal component.

To obtain the equations for the longitudinal and transverse variations

of the ﬁeld amplitudes, we assume a Gaussian incident light beam

E = E

−r

= x

+ y

with a characteristic (transverse) width w

which propagates in z direction.

Using the ansatz

∇ = e

∇

+ e

∂

∂z

E = e

−i(ωt−kz)

+ e

)

P = e

−i(ωt−kz)

+ e

) , (16.2)

we subdivide the wave equation (16.1) into a longitudinal and a transverse

part. e

and e

in Eq. (16.2) are the unit vectors in z-direction and in the

transverse directions, respectively. The transverse equation is

∇



∇



ik +

∂

∂z





−



−k

+2ik

∂

∂z

∂

∂z

+ ∇





+2iω

∂

∂t

−

∂

∂t



+4πP

, (16.3)

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Optical Bistability 307

and the longitudinal equation is



ik +

∂

∂z



∇

−∇



+ i2ω

∂

∂t

−

∂

∂t



+4πP

. (16.4)

As usual, the polarization is related to the optical susceptibility or the

dielectric function via

z,T

= χE

z,T

 − 1

4π

z,T

. (16.5)

It is now convenient to split the susceptibility into the linear part χ

and

the nonlinear, density-dependent part χ

χ = χ

+ χ

(N)=



− 1

4π



(N)

4π

. (16.6)

Even though we explicitly deal with a density-dependent nonlinearity, this

treatment can easily be generalized to also include, e.g., thermal and other

nonlinearities. To derive the coupled equations for the longitudinal and

transverse ﬁeld components from Eqs. (16.3) and (16.4), we use the so-

called paraxial approximation, see Lax et al. (1975). Scaled variables are

introduced as

x =¯x w

; y =¯y w

; z =¯z l ; t =

t τ

, (16.7)

where

l = kw

(16.8)

is the diﬀraction length of the beam in z direction, τ

= ln

/c is the char-

acteristic propagation time over that distance, n

√



and ω = ck/n

The dimensionless number

f =

<< 1 (16.9)

serves as small parameter allowing us to expand all quantities in powers of

f. Consistent equations are obtained using

= E

[0]

+ f

[2]

+ ··· ,

= fE

[1]

+ f

[3]

+ ··· ,

= f

[2]

+ f

[4]

+ ··· . (16.10)

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308 Quantum Theory of the Optical and Electronic Properties of Semiconductors

Inserting Eqs. (16.9) and (16.10) into Eqs. (16.3) and (16.4) yields a set of

coupled equations in the orders O(f), O(f

), ···. Restricting ourselves to

O(f ),weobtain

[1]

= i ∇

[0]

(16.11)

and



∂

− i

2kn

∇

∂

∂t

cα(ω, N)

− i

ω∆n(ω, N)



[0]

=0 ,

(16.12)

transverse ﬁeld equation

where we used Eqs. (1.51), (1.50) to introduce absorption and refractive-

index change,

α(ω,N)=

4πω



(ω,N) and ∆n(ω) 

2πχ



(ω,N)

, (16.13)

respectively. Eq. (16.10) shows that the electromagnetic ﬁeld is purely

transverse in lowest order, but it may nevertheless depend on the transverse

coordinate. Only in next order, there is a small longitudinal component

whosesizedependsonf.

For simplicity of notation, we now replace e

[0]

by E. Through

α(ω,N) and ∆n(ω, N), the ﬁeld amplitude E is nonlinearly coupled to

the electron–hole–pair density

N =



e,k



h,k

. (16.14)

Note, that we use N for the electron–hole–pair density in this chapter to

distinguish it clearly from the refractive index n.