For doped semiconductor superlattices we have analyzed the effects of absorption saturation and
change in the refractive index depending on the level of their excitation and structure parameters.
The calculations were carried out with account for the tails of the density of states and screening of
the electrostatic potential. It is shown that doped superlattices may display "shading," i.e., the increase
in the absorption coefficient at a fixed frequency with increase in light intensity. For δ-doped
superlattices a stronger nonmonotonicity of the change in the refractive index with increase in the
excitation level in comparison with typical structures can manifest itself

Kononenko, V. K.

Manak, I. S.

Ushakov, D. V.

English

Электронная библиотека БГУ

NONLINEAR OPTICAL PROCESSES IN DOPED
SEMICONDUCTOR SUPERLATTICES*
D. V. Ushakov,a** V. K. Kononenko,b
and I. S. Manaka
UDC 539.293:621.382
For doped semiconductor superlattices we have analyzed the effects of absorption saturation and
change in the refractive index depending on the level of their excitation and structure parameters.
The calculations were carried out with account for the tails of the density of states and screening of
the electrostatic potential. It is shown that doped superlattices may display "shading," i.e., the in-
crease in the absorption coefficient at a fixed frequency with increase in light intensity. For δ-doped
superlattices a stronger nonmonotonicity of the change in the refractive index with increase in the
excitation level in comparison with typical structures can manifest itself.
Keywords: doped superlattice, absorption saturation, nonlinearity parameter, refractive index.
Semiconductor structures on doped superlattices have attracted attention because of the widespread
possibilities of rearranging their characteristics on both a change in the constructional-technological parame-
ters of the superlattice and structure excitation. Alternate doping of a semiconductor with donor and acceptor
impurities leads to modulation of the edges of the conduction and valence bands, resulting in a decrease in
the effective energy gap width Eg′ , and periodically recurring spatially separated potential wells are formed for
electrons and holes. The depth and profile of the potential wells are determined by the thickness of the layers
of n- and p-type, dn and dp, by the concentration of the donors and acceptors Nd and Na, by the thickness of
undoped i layers di, and also by the two-dimensional concentrations of nonequilibrium electrons n and holes
p [1].
On excitation of a doped superlattice, the charge of the donor and acceptor impurities is screened by
nonequilibrium current carriers, and this causes a decrease in the depth of the potential wells and, correspond-
ingly, an increase in the effective energy gap width. The dependence of Eg′  on the excitation level of the
structure is responsible for the observed shift to the region of higher energies of the longwave edge of the
absorption and luminescence spectra of doped GaAs-based superlattices [2–5].
The change in the properties of doped superlattices with the concentration of nonequilibrium current
carriers leads to nonlinear optical effects. In particular, at a given light frequency ν we observe the depend-
ence of the absorption coefficient k(ν) and refractive index nr(ν) on the intensity of light [6, 7]. The absorp-
 
*Reported at the 3rd International Scientific-Engineering Conference on Quantum Electronics (Minsk, November 20–22,
2000). 
**To whom correspondence should be addressed.
aBelarusian State University, 4 F. Skorina Ave., Minsk, 220050, Belarus; e-mail: Usha-kovDV@
rfe.bsu.unibel.by; bB. I. Stepanov Institute of Physics, National Academy of Sciences of Belarus, Minsk, Be-
larus. Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 68, No. 4, pp. 501–505, July–August, 2001.
Original article submitted February 3, 2001.
Journal of Applied Spectroscopy, Vol. 68, No. 4, 2001
0021-9037/01/6804-0656$25.00 2001 Plenum Publishing Corporation656
tion and refraction spectra can rearrange on both optical excitation and application of electric voltage to the
structure. In the present work, we consider the characteristic features of the change in the absorption coeffi-
cient and refractive index in a doped superlattice depending on the density of exciting radiation.
Absorption Saturation. As is known, saturation of the absorption power or light amplification is as-
sociated with a decrease in the absorption coefficient or amplification of light at a fixed frequency ν with
increase in the amplitude of the electromagnetic field strength. In semiconductor systems this effect is caused
by the approach of the difference of the Fermi quasilevels ∆F to the energy of photons hν on increase in the
intensity of monochromatic radiation. The characteristic features of nonlinear absorption or amplification of
light were analyzed earlier for bulk semiconductors [8–10] and quantum-dimensional systems [11–13]. For
doped semiconductor superlattices this effect has not been studied fully [6, 7, 14]. Moreover, the effect of
fluctuations of the electrostatic potential on linear optical processes has not been investigated.
The stationary equation that determines the coupling between the absorption or amplification coeffi-
cient k and the density of photons S has the form [8]
η′j
ed  = 
Rlum
ηsp
 % νgk (ν) S , (1)
where j is the current density per period of the superlattice d, η′ is the injection efficiency, ηsp is the quan-
tum yield of luminescence, and νg is the light velocity in a crystal. In Eq. (1) the second term has the
"minus" sign for absorption processes and the "plus" sign for light amplification. The rate of radiative recom-
bination Rlum is determined by integration of the velocity of optical transitions rlum(hν) over all the energies
of photons. The quantity rlum(hν) is related to the rate of spontaneous recombination rsp(hν) at a given tem-
perature T and level of excitation of the crystal ∆F by the relation [9]
rlum (hν) = 
1 − exp − 
∆F
kT

1 − exp − 
hν
kT

 rsp (hν) .
The general expression for the rate of spontaneous recombination in the case of direct transitions has the
form [1]
rsp (hν) = 
Acν
π²
2Npd
  ∑ 
i
 mriM  ∑ 
n
 ∑ 
m
 ∑ 
ν
 Ht (hν − hνnmiν) Inmiν2 fe (Ecnmiν) fh (Eνnmiν) , (2)
where Acν is the Einstein coefficient, Np is the number of periods of the superlattice, Inmiν are the overlap
integrals of the envelope wave functions of electrons and holes, and fe(Ecnmiν) and fh(Evnmiν) are the Fermi–
Dirac distribution functions for electrons and holes with the energies
Ecnmiν = Ec0 + 
mriM
mc
 
hν − Eg′  + 
mriM
mνiM
 Ecnν − 
mriM
mc
 Eνinν ,
Eνnmiν = Eν0 − 
mriM
mνiM
 
hν − Eg′  + 
mriM
mνiM
 Ecnν − 
mriM
mc
 Eνimν ,
Ec0 and Eν0 are the energies of the conduction band bottom and of the valence band top, mc, mνhM, and
mνlM are the effective mass of the electrons and the transverse components of the effective masses of heavy
657
and light holes, respectively, and mriM = mcmνiM/(mc + mνiM) is the reduced mass. For GaAs (with the plane
orientation {100}) we use mc = 0.067me, mνhM = 0.11me, and mνlM = 0.20me. Summation in (2) is carried out
over the quantum numbers of the subbands of electrons n, holes m, and minisubbands ν, and over the states
of heavy and light holes (i = h, l). Transitions between the subbands begin from the energies of the quanta
of light hνnmiν = Eg′  + Ecnν + Evimν, associated with the effective energy gap width of the doped superlattice
Eg′  = Ec0 − Eν0 and with the energies of the levels of the subbands of electrons Ecnν and holes Evimν.
The function Hi(y) is governed by the broadening of the energy spectrum. In the absence of broaden-
ing effects in a semiconductor structure, Ht(y) represents the Heaviside step function. For hightly doped su-
perlattices the impurity and the proper zones of a crystal overlap; therefore the tails of the density of states
appear [1]. In the case of the Gaussian tail of the density of states, the function Ht(y) has the form Ht(y) =
erfc (−y/σcv)/2 [5], where σcv = (σc2 + σv2)1/2 and σc and σv are the characteristic parameters of the tails of the
density of states in the conduction and valence bands. With increase in the level of excitation of the doped
superlattice, the tails of the density of states become shorter due to the screening of the fluctuating impurity
potential by nonequilibrium current carriers [1, 15].
The absorption coefficient k(ν) is related to the rate of spontaneous recombination rsp(hν) by the uni-
versal relation [9]
k (ν) = 
exp 
hν − ∆F
kT
 − 1
νg ρ (hν)
 
rsp (hν) ,
(3)
where ρ(hν) = (hν)2nr2/π2c2²3νg is the density of electromagnetic modes and nr is the refractive index of the
crystal. Using expressions (1)–(3), we find the dependence of the absorption coefficient k on the density of
photons S. In the general case it obeys a complex law [8]. If the mean value of the nonlinearity parameter
α is taken into account, then the change in the absorption (amplification) coefficient can be described by the
simple formula [8, 9]
k = 
k0
1 + hναS
 , (4)
in which k0 is the initial coefficient of absorption (amplification).
To carry out a more detailed analysis of the effects of saturation in specific quantum-dimensional
structures, numerical calculations are needed. For doped semiconductor superlattices it is convenient to per-
form the calculation of nonlinear absorption by the following algorithm [13]. First, by prescribing the differ-
ence of the Fermi quasilevels ∆F0 and assuming the density of photons S = 0, it is necessary to determine
the corresponding coefficient of absorption k0 at the exciting light frequency ν and the current density j. If j
= 0, then ∆F0 = 0. Increasing ∆F, we find new values of the absorption coefficient k and the recombination
rate Rlum, and thereafter the corresponding density of photons S is calculated from Eq. (1). Thereupon the
procedure is repeated until the difference of the Fermi quasilevels ∆F becomes equal to the energy of photons
hν.
The results of the calculations of absorption saturation for two types of doped GaAs-based superlat-
tices for j = 0, T = 300 K, and ηsp = 0.7 are presented in Fig. 1. The calculations of k(ν) and rsp(hν) were
made in the model of the Gaussian tails of the density of states with account for the screening of the fluctu-
ating impurity potential. As is seen, the absorption coefficient k at a fixed frequency ν exhibits nonmonotonic
behavior on increase in the density of photons S. The "shading" effect, i.e., the increase in the absorption
coefficient k on increase in the density of photons S, is due to the transformation of the potential energy
profile of the doped superlattice, redistribution of the energy levels in quantum wells for electrons and holes,
658
and change in the overlap integrals of wave functions of electrons and holes, and also to the narrowing of the
forbidden band.
In the absence of fluctuations of the concentrations of impurities an oscillative change in the absorp-
tion coefficient at a fixed frequency of light is observed, depending on the level of excitation of the doped
superlattice [6]. In doped superlattices, the fluctuations of the electrostatic potential usually smooth out the
stepwise density of states, which leads to smooth absorption and emission spectra [1]. As a result, the func-
tion k(S) also becomes smooth, but the region of shading, which most often displays itself at the energy of
the photons of the order of the energy gap width of the semiconductor, does not disappear (Fig. 1).
The nonmonotonic character of the absorption saturation is most pronounced for the δ-doped super-
lattice (Fig. 1b). For bulk semiconductors the shading can be caused by the increase in the absorption by free
carriers [8]. However, this process practically does not influence the absorption saturation in doped superlat-
tices, since the coefficient of light absorption by free carriers for the energies of photons of the order of the
effective energy gap width of the superlattice does not exceed 5 cm−1 [16].
In a wide range of change in the density of photons S, the function k(S) is not described by formula
(4) with a constant nonlinearity parameter α. It is more convenient to carry out evaluations of the parameter
α using a double logarithmic scale [8] for the dependence of k0/k − 1 on S (Fig. 2). Then the behavior of the
curves will show the change in α at the different energies of exciting quanta hν depending on the range of
the light flux density hνvgS.
As is seen, the threshold bleaching fluxes, when the absorption coefficient decreases, e.g., by 1%, for
a doped superlattice of p-type is D10 mW/cm2 for hν = 0.8 eV. With increase in hν the bleaching thresholds
become still smaller. For a δ-doped compensated superlattice the opposite picture is observed: as hν in-
creases, the bleaching threshold rises, and at hν = 1.4 eV it comes to about 2 kW/cm2. The difference is due
to the different character of filling of the states in the conduction and valence bands in optical excitation of
superlattices.
As the density of photons S increases, the nonlinearity parameter decreases at any energy of light
quanta. This change is attributable to the joint effect of the shortening of the tails of the density of states [10]
and to the increase in the effective energy gap width [1]. The latter factor predominates for δ-doped compen-
sated superlattices because of the strong background screening [5].
Fig. 1. Absorption saturation for different types of doped superlattices at
different energies of photons: a) Na = 1019 cm−3, Nd = 6⋅1018 cm−3, dp =
35 nm, dn = 25 nm, and di = 0; b) Na = Nd = 1020 cm−3, dp = dn = 1
nm, di = 8 nm; 1) hν = 0.8 eV, k0 = 0.1 (a) and 0.4 cm−1 (b); 2) hν =
1.1 eV, k0 = 20 (a) and 84 cm−1 (b); 3) hν = 1.4 eV, k0 = 1537 (a) and
2437 cm−1 (b).
659
For large enough values of S, the change in the absorption coefficient follows formula (4) with a
constant value of α [8]. In particular, for the energies of photons hν = 0.8, 1.1, and 1.4 eV we respectively
have α/vg = 0.83, 0.26, and 0.010 cm
2/kW for the p-type superlattice (Fig. 2a). For the same energies of
photons in the case of a δ-doped superlattice we obtain α/vg = 0.10, 0.028, and 0.0067 cm2/kW (Fig. 2b). As
is seen, the values of α decrease with increase in hν. The light flux density at which the absorption coeffi-
cient becomes two times as small varies within wide ranges: from 2W/cm2 in the p-type superlattice to 80
kW/cm2 in the δ-doped superlattice.
Nonlinear Dispersion. As is shown in [7], in doped superlattices there is a stronger nonlinear refrac-
tion in comparison with bulk semiconductors. The difference in the refractive indices of a doped GaAs-based
superlattice and a bulk semiconductor is equal to 0.02. The difference between the refractive indices in ex-
cited and unexcited states of doped superlattices based on GaAs attains 0.003 and those based on InAs 0.015.
In δ-doped superlattices, still greater nonlinear refraction is possible [17].
The dispersion characteristics of the δ-doped GaAs-based superlattice were calculated with account
for screening of the electrostatic potential by current carriers and of the Gaussian tails of the density of states
[5]. The refractive index ∆nr at a certain frequency ν0 was determined from the Kramers–Kronig relation
∆nr (ν0) = c2π2
  
k
 (ν)
ν
2
 − ν0
2 dν .
Here, the integral is the principal value.
The results of calculations of the absorption coefficient k(ν) and refractive index ∆nr(ν) are presented
in Fig. 3. As is seen, with increase in the excitation level ∆F the maximum of the dispersion curve is shifted
to the shortwave region by 100 meV, which corresponds to the shift of the absorption edge because of the
increase in the effective energy gap width of the superlattice.
The dependences of the refractive index ∆nr at a fixed frequency of light on the difference of the
Fermi quasilevels ∆F are presented in Fig. 4. For the δ-doped superlattice the nonmonotonic character of the
change in ∆nr is a stronger function of ∆F, especially for the energies of photons of the order of the energy
gap width of the crystal. Here, the change in the refractive index at a fixed frequency with a change in the
excitation level of the superlattice for T = 300 K can attain 0.02.
Using relation (1), one can easily determine the dependence of ∆nr on the density of exciting photons
S. At rather large values of hν the nonmonotonic character of the nonlinear refraction ∆nr(S) is clearly dis-
Fig. 2. Dependence of k0/k − 1 on the density of photons S at different
energies of photons for doped superlattices of different types. The pa-
rameters of the superlattices are the same as in Fig. 1a and b, curves
1–3, respectively.
660
played. In the limiting case, ∆nr seems to approach a constant value that corresponds to the condition ∆F =
hν [8, 13].
The quantity ∆nr should be considered as the addition to the refractive index of the crystal due to the
dimensional quantization of levels and the filling of them with nonequilibrium current carriers on excitation
of the superlattice. The rather large changes in the absorption coefficient and refractive index in the semicon-
ductor structures with a doped superlattice can be utilized in optical bi- and multistable switches [18]. More-
over, these structures can be built into photon crystals for tuning transmission resonances with a view toward
developing low-threshold optical logical elements [19].
This work was carried out within the framework of Belarus-Russia cooperation on a project of the
Belarus Republic Basic Research Foundation (No. F99R-119).
Fig. 3. The spectra of absorption k(ν) (a) and refraction ∆nr(ν) (b) for
the δ-doped superlattice at different excitation levels: ∆F = 0.8 (1), 1.0
(2), 1.2 (3), and 1.4 eV (4). The parameters of the superlattice are the
same as in Fig. 1b.
Fig. 4. Change in the refractive index ∆nr at a fixed frequency ν de-
pending on the difference between the Fermi quasilevels ∆F: hν = 0.8
(1), 0.9 (2), 1.0 (3), 1.1 (4), 1.2 (5), 1.3 (6), and 1.4 eV (7). The pa-
rameters of the superlattices are the same as in Fig. 1a and b, respec-
tively.
661
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Nonlinear optical processes in doped semiconductor superlattices

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Nonlinear optical processes in doped semiconductor superlattices

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