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Effect of Coulomb interaction on nonlinear (intensity-dependent) opticalprocesses and intrinsic bistability in a quantum well under the electric andmagnetic fields

_Ibrahim Karabuluta)

 Department of Physics, Selcuk University, Konya 42075, Turkey

(Received 7 December 2010; accepted 27 December 2010; published online 2 March 2011)The effect of the electrostatic Coulomb interaction on the line shape related to the

intensity-dependent intersubband optical processes in a Si d-doped quantum well is studied using

the density matrix formalism. The electronic structure of the quantum well is calculated from the

self-consistent numerical solution of the coupled Schrodinger–Poisson equations. The line shape

function is considerably modified by the optical intensity and the electric and magnetic fields.

Moreover, we demonstrate the existence of the optical bistability for appropriate values of the

optical intensity and also the control of the optical bistability with the electric and magnetic fields.

It is also found that the intersubband relaxation time plays an important role in determining the

optical bistability region.VC 2011 American Institute of Physics. [doi:10.1063/1.3553641]

I. INTRODUCTION

In recent years, intersubband optics has become a topic

of increasing interest due to recent advances in quantum cas-

cade lasers and terahertz (THz) devices. Intersubband transi-

tions (ISBTs) in semiconductor quantum wells (QWs)

exhibit large oscillator strengths and take place on a meV

energy scale. Therefore, large optical nonlinearities related

to these transitions have been obtained.1,2

At low electron densities usually encountered in semi-

conductor QWs, the electron–electron (e-e) interaction is

negligible, and so the single electron approach becomes a

valid approximation. This electron satisfies the single-parti-

cle, linear Schrodinger equation and resonantly absorbs lightat a frequency equal to the energy difference between ground

and first excited states. Thus for a single-electron case, the

spectral response is nearly Lorentzian. It is also well known

that many electrons are present in real QWs, and the e-e

interaction plays an important role especially in heavily

delta-doped GaAs QWs.3 – 5 A single electron approximation

fails in modeling and interpretation of the experimental data

on the intersubband optical properties in semiconductor 

QWs with high electron concentration, and so it is necessary

to take into account the effects of e-e interaction on the cal-

culation of intersubband optical processes. This study is

aimed at clarifying the contribution of e-e interaction to the

line shape related to intensity-dependent optical processessuch as nonlinear optical absorption (NOA) and nonlinear 

optical rectification (NOR) in a Si d-doped QW under the

electric and magnetic fields.

Many studies regarding the many-body effects on the

intersubband transitions and the optical properties of QWs

have been carried out in recent years.3 – 10 Most of these have

been associated with the dynamic screening (that is, depola-

rization) effect.4 – 10

Another important effect resulting f rome-e interaction is the electrostatic Coulomb interaction.11,12

As compared with the depolarization effect, this effect has

been less studied in the literature. It is also well known that

these effects tend to cancel each other.12 The asymmetry of 

the potential profile is very important for nonlinear optical

applications. It is possible to adjust the asymmetry of the

potential profile by applying electric and magnetic fields.

Moreover, we have seen from our numerical calculations

that both depolarization and electrostatic Coulomb effects

are directly related to the asymmetry of the potential profile.

Especially in strongly asymmetric QWs, the electrostatic

Coulomb effect dominates the depolarization effect. In sub-

sequent calculations, we will focus on this case and ignore

the contribution of the depolarization effect to intersubband

line shape.

II. THEORY

In this work, we have used the perturbative approach

developed using rotating wave approximation (RWA) by

Zaluzny.12 It is well known that this theory is capable of 

explaining the experimental data.6 The method used in the

present calculations is based on the density matrix formal-

ism, and it has already been given elsewhere, and we willnot enter into details here.12 The expression for the line

shape related to the intensity-dependent NOA and NOR

coefficients is given by

Gðx; I Þ ¼~C

22

d2 þ ~C22

1

1 þ I = I 0; (1)

where d ¼ hx À e21ð1 À ceDÞ, Ci ¼ hT i (i ¼ 1, 2), ~C22

¼ C22ð1 þ I = I 0Þ, Z 0 is the vacuum impedance (120 pX), I  is

the intensity of the optical radiation inside the QW, and

 I 0 ¼ nC1C2=2e2 /1 zj j/2ihj j2 Z 0 is the optical saturation

a)Author to whom correspondence should be addressed. Electronic mail:

ikarabulut@selcuk.edu.tr.

0021-8979/2011/109(5)/053101/6/$30.00 VC 2011 American Institute of Physics109, 053101-1

JOURNAL OF APPLIED PHYSICS 109, 053101 (2011)

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intensity. D is the expression for the population change and

is determined through

D ¼ÀC2

2

e21ð1 À ceDÞ À hx½ �2þC22ð1 þ I = I 0Þ

 I 

 I 0: (2)

Also ce parameter is the maximum shift of the intersubband

energy expressed in the unit e21 and is given by

ce ¼ 0:7 Â ðjGaAs=jÞð N ð1ÞS À N 

ð2ÞS Þð1012 cmÀ2Þ

 hdzieðAÞ=e21ðmeVÞ; (3)

with

dzh ie¼

ð 1À1

dz

ð zÀ1

dz0 /1ðz0Þj j2

À /2ðz0Þj j2

h i 2

(4)

 N ðnÞs is the electron concentration in the nth subband.

Looking at the expressions in the preceding text, it is not

hard to see that the electrostatic Coulomb interaction directly

depends on ce

and D. Thus to correctly calculate the influ-

ence of this interaction, ce and D should also be calculated

exactly. Therefore, one has to determine the electronic struc-

ture of the system by self-consistent solution of Schro-

dinger–Poisson equations. Unfortunately, in most of the

studies in the literature, this procedure has been omitted, and

a constant value for  ce has usually been used directly. How-

ever, in this study, we solve self-consistently and numeri-

cally the coupled Schrodinger–Poisson equations and

calculate ce values. In addition, in our calculations, the

effects of the electric and magnetic fields are also taken into

account to demonstrate controllability of the optical

bistability.

In this paper, a single Si d-doped GaAs layer embeddedin the center of an infinite QW with the thickness of  L is con-

sidered. The delta doping of Si atoms in GaAs QW is an effi-

cient way to obtain the quantum heterostructures with a high

density of carriers.13 Such quantum structures ar e very

attractive for infrared devices based on the ISBTs.13,14

In the effective mass approximation, the one-dimen-

sional Schrodinger equation for a Si d-doped GaAs QW

subjected to a crossed electric field and magnetic field is

given by

Àh2

2mÃ

d 2

dz2þ V conf ðzÞ þ V  H ðzÞ þ V  XCðzÞ þ eEz

þ

1

2mÃx2

c z À z0ð Þ2

!/iðzÞ ¼ ei À

h2k 2 y

2mÃ

" #/iðzÞ: (5)

Here, mà is the effective mass, V conf ðzÞ is the built-in poten-

tial due to the infinite barriers, E is the applied electric field

strength along the growth direction [¼(0,0, E)], xc

¼ eB=ðmÃcÞ is the cyclotron frequency, z0 ¼ Àhk  x=eB is the

Landau level center coordinate, B is the applied magnetic

field strength perpendicular to the growth direction [A¼ (zB,

0, 0)], /iðzÞ is the envelope wave functions, and ei is the

energy eigenvalues.

The Hartree potential [V  H ðzÞ], stems from the electro-

static interaction of the electrons with themselves and with

ionized dopants is obtained from the solution of one-dimen-

sional Poisson equation:

d 2V  H ðzÞ

dz2¼

4pe2

jN 3 D

 D ðzÞ À nðzÞÀ Á

; (6)

with the electron density is given by

nðzÞ ¼X

i

ni /iðzÞj j2; (7)

and

ni ¼mÃk  BT 

p h2ln 1 þ exp

E F À ei

k  BT 

!; (8)

where ni is the number of electrons per unit area in the ith

subband, j is the local dielectric constant, and N 3 D D ðzÞ is the

total density of ionized dopants. E F is the Fermi energy and

is obtained from the solution of 

 N 

2 D

 D ¼X

i ni; (9)

where N 2 D D is the 2 D donor doping density. Using the para-

metrization suggested by Hedin and Lundqvist, the

exchange-correlation potential V  XCðzÞ within the local-den-

sity approximation (LDA) is given by15

V  XCðzÞ ¼ ÀRyÃ

10:5pa r ½1 þ 0:7734r lnð1 þ r À1Þ�; (10)

where a ¼ 49p

À Á1=3, r  ¼ 1

2143p aÃ3nðzÞ

À ÁÀ1=3, Ryà ¼ e2=2j aÃ

is the effective Rydberg and aà ¼ jh2=mÃe2 is the effective

Bohr radius.

III. NUMERICAL RESULTS AND DISCUSSION

The electronic structure of such a system discussed in

the preceding text has to be determined by the self-consistent

solution of the coupled Schrodinger–Poisson equations.

Without loss of generality, and for simplicity in numerical

calculations, we have set fk  x; k  yg ¼ 0. We have performed

numerical calculations for the Si d-doped GaAs layer with

 L ¼ 50 nm (the quantum well width) and N 2 D D ¼ 5 Â 1012

cmÀ2. The calculations have been done for the temperature

4.2 K. We assume that the donor atoms are distributed homo-

genously in a sheet of thickness Dz so that N 3 D D ¼ N 2 D

 D =Dz.

The thickness of the Si d-doped layers is assumed to be2 nm. In our calculations, the effective mass of the electron

is taken as mà ¼ 0:067m0, m0 being the free electron

mass, and j ¼ 12:7, and the relaxation times are taken as

T 1 ¼ 1 ps, T 2 ¼0.2 ps.1 Because the density of electron in a

Si d-doped QW exceeds 1011 cm-2, the Hartree ter m com-

pletely dominates the exchange-correlation term.11,12 There-

fore, the effects related to exchange-correlation interaction

are ignored in our calculations.

In Fig. 1, the effects of the optical intensity and the elec-

tric and magnetic fields on the line shape function Gðx; I Þare shown, respectively. We can see from Fig. 1(a) that in

the low saturation limit ( I  ( I 0), the line shape function

Gðx; I Þ is nearly Lorentzian. When the intensity of the light

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reaches a value comparable to the saturation intensity I 0, the

peak of Gðx; I Þ shifts to higher energies by an amount

depending on the intensity of the light (here, I 0 ffi 0.14 MW/ 

cm2). For the intensity values larger than I 0, the peak of 

Gðx; I Þ shifts more than tails and the line shape of the corre-

sponding function becomes more asymmetric.

It is also interesting to note that the line shape function

Gðx; I Þ has the bistable characteristic at high intensities.

In this bistable region, the line shape function Gðx; I Þ can

switch suddenly between two stable solutions. This case

obtained at high intensities is not observed in the study of 

Zaluzny (see Fig. 1).12 This is because the value of  ce

(%0.413) parameter obtained in this model is larger than that

of ce (¼0.1) used by Zaluzny. We also want to stress that the

theory used in our calculations breaks down when the shift

of intersubband energy arising from the electrostatic

FIG. 1. (Color online) Line shape function Gðx; I Þ as a function of the photon energy. In (a) E ¼ 40 kV/cm, B ¼ 0, (b) I ¼ 0.07 MW/cm2

, B ¼ 0, and (c)

 I ¼ 0.07 MW/cm2

, E ¼ 40 kV/cm.

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Coulomb interaction reaches a value comparable to e21, and

so the present theory should be improved at this point. From

Eq. (4), it is clear that the ce parameter is proportional to the

spatial separation of the ground and the first excited state

wave functions. Moreover, it can be seen that this separation

increases with the asymmetry of the potential profile. There-

fore, it is possible to adjust the asymmetry and ce parameter 

and so the optical bistability by applying the electric and

magnetic fields. For this purpose, in Figs. 1(b) and 1(c), the

effects of the electric and magnetic fields on Gðx; I Þ are

shown. The increase in the electric field leads to an increase

in ce and to the appearance of the optical bistability in Si d-

doped GaAs QW. This result can be attributed to the fact

that the asymmetry of the QW potential increases with the

applied electric field. Moreover, the peak value of Gðx; I Þ

increases and the peak position shifts toward blue with the

FIG. 2. (Color online) Line shape function Gðx; I Þ as a function of the optical intensity of the incident light. In (a) E ¼ 40 kV/cm, B ¼ 0, (b) hx ¼ e21 þ 3C2,

 B ¼ 0, and (c) hx ¼ e21 þ 3C2, E ¼ 40 kV/cm.

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applied electric field. The peak value and position of Gðx; I Þare equal to 1=ð1 þ I = I 0Þ and hx ¼ e21ð1 þ ce I = I 0=ð1

þ I = I 0ÞÞ, respectively. It can be easily seen that the peak

position increases with both ce and I . This is the reason for 

the blueshift of line shape with the applied electric field and

optical intensity. Moreover, from our numerical calculations,

an increase in I 0 resulting from the decrease in /1h jz /2j ij j2

is observed with the applied electric field. This increase leadsalso to an increase in the peak value of Gðx; I Þ. The effect

of the magnetic field on Gðx; I Þ is very different from the

effect of the electric field on it. First, the asymmetry of the

potential profile decreases with the applied magnetic field.

This decrease in the asymmetry shows itself with a decrease

in ce and so leads to disappearance of the optical bistability.

Optical bistability is one of the most active research areas in

intersubband optics.16 – 19 Therefore, the control of both the

optical bistability and the peak value and position of G ðx; I Þwith the electric and magnetic fields is very important for 

nonlinear optical applications.

In Fig. 2(a) the line shape function Gðx; I Þ as a function

of the optical intensity of light for four different values of the photon energy are shown. It is easily seen from Fig. 2(a)

that for larger photon energies, the intensity dependence of 

Gðx; I Þ exhibits clearly the hysteresis loops in a certain

range of the light intensity, and in this intensity region,

Gðx; I Þ is bistable. The intensity range, where the hysteresis

loop is observed, increases dramatically with the photon

energy. It is also noted that the peak value of Gðx; I Þdecreases with the photon energy. Figs. 2(b) and 2(c) show

Gðx; I Þ as a function of light intensity for different values of 

the electric and magnetic fields. It is noted from Fig. 2(b)

that in the absence of the electric field, the characteristic S

curve implying the existence of bistability disappears. It is

obvious from Figs. 2(b) and 2(c) that while the increase in

the electric field leads to a decrease in the threshold of the

onset of the optical bistability, the increase in the magnetic

field increases this threshold value. Also it is clearly shown

that with increasing values of the applied electric field, the

area of the hysteresis loop becomes narrower; on the other 

hand, this area becomes wider with the magnetic field.

We now consider the case in which there are multiple

solutions for the population change implying the optical bist-

ability. Eq. (2) can be written as

D ½ ðd þ deDÞ2 þ 1 þ x� þ x ¼ 0; (11)

where d ¼ ðhx À e21Þ=C2 is the detuning parameter,

de ¼ e21ce=C2, and x ¼ I = I 0. It is possible to rewrite Eq. (11)

in the form

D3 þ a2D

2 þ a1D þ a0 ¼ 0; (12)

where a0 ¼ x=d2e , a1 ¼ ð1 þ d2 þ xÞ=de, and a2 ¼ 2d=d2

e .

Then we look for the sign of the parameter 

 D ¼ q3 þ r 2; (13)

where q ¼ a1=3 À a22=9, and r  ¼ ða1a2 À 3a0Þ=6 À a3

2=27.

To obtain multiple physical solutions, a necessary condition

is D 0.20

Finally, it is instructive to investigate the ranges of  d

and x for which multiple physical solutions exist. The results

obtained are given in Fig. 3 for three different values of 

relaxation time T 2. The area inside the closed curves corre-

sponds to multiple physical solutions of D implying bistabil-

ity, whereas the outside area of the closed curves

corresponds to a single physical solution for D. From our nu-

merical calculations, given in Fig. 3, the increase in T 2 leads

to an enlargement of multiple solutions region. Thus the

intersubband relaxation times play a key role in determining

the optical bistability region.

IV. CONCLUSIONS

In summary, the effects of e-e interaction on the line

shape function Gðx; I Þ in a Si d-doped QW are studied. It is

found that the electrostatic Coulomb interaction shifts the

peak position of Gðx; I Þ to blue and leads to the significant

asymmetry of the line shape. In addition, the optical bistabil-

ity can be easily controlled by the photon energy and the

electric and magnetic fields. Although the theory used in this

study is well matched with the experimental results, a more

complete theory including the variation of the relaxation

time with intensity and also the contribution of LO phonon is

needed. We hope that the present study will be useful in the

design of the devices based on the intensity-dependent opti-cal processes.

ACKNOWLEDGMENTS

_I. K. wishes to thank U. Atav, I. Carusotto, and

M. Zaluzny for useful discussions. This work was partially

supported by the BAP office of Selcuk University.

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FIG. 3. (Color online) The ranges of values of the detuning parameter d and

the intensity of the optical radiation x for which optical bistability is avail-

able (marked with the closed curves).

053101-5 _Ibrahim Karabulut J. Appl. Phys. 109, 053101 (2011)

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