Available online at http://www.idealibrary.com ondoi:10.1006/spmi.2001.0975Superlattices and Microstructures, Vol. 29, No. 4, 2001

Pump–probe absorption/gain spectroscopy of strongly confinedsemiconductor nanocrystals

PRANAY K. SEN†

Department of Applied Physics, Shri G S Institute of Technology and Science, Indore-452003, India

J. THOMAS ANDREWS‡

Department of Applied Physics, Birla Institute of Technology, Mesra, Ranchi-835215, India

(Received 22 November 2000)

The density matrix approach has been employed to analyze the pump–probe spectroscopicabsorption spectra of small semiconductor nanocrystals popularly known as quantum dots(QDs) under the strong confinement regime (SCR) with sizes smaller than the bulk excitonBohr radius such that the Coulombic interaction energy becomes negligible in comparisonwith the confinement energy. The average time rate of absorption has been obtained byincorporating the radiative and nonradiative decay processes as well as the inhomogeneousbroadening arising due to nonuniform QD sizes. The analytical results are obtained for QDsduly irradiated by a strong near-resonant pump and a broadband weak probe. Numericalestimations have been made for: (i) isolated QDs and (ii) QD-arrays of GaAs and CdS.The results agree very well with the available experimental observations in CdS QDs. Theresults in the case of GaAs QDs can lead one to experimentally estimate absorption/gainspectra in the important III–V semiconducting microscopic structures.

c© 2001 Academic PressKey words: quantum dot, absorption spectra, biexcitons, inhomogeneous size distribution,pump–probe technique.

1. Introduction

The last quarter of the twentieth century has witnessed a tremendous growth in the study of coherentradiation–matter interactions in semiconductors. Recent advances in the synthesis of zero-dimensional semi-conductor nanocrystals known as quantum dots (QDs) have attracted the attention of both theorists andexperimentalists, due to the giant optical nonlinearity and flexible modifications in electronic and opticalproperties through structure designs [1, 2]. The recent crystal growth technology of QDs has made it possi-ble to devise room-temperature single-electron memory elements for use in fast optoelectronic devices [3].Measurement of the nonlinear response of such QDs indicates favorable characteristics in terms of switch-ing speed, magnitude of nonlinearity and switching energy; establishing them as the promising candidatematerials for the development of photonic devices for optical communication, switching and computing [4].†E-mail:pratima_sen@hotmail.com‡Present address: Biomedical Applications Section, Center for Advanced Technology, Indore-452013, India. E-mail:jta@cat.ernet.in

0749–6036/01/040287 + 13 $35.00/0 c© 2001 Academic Press

288 Superlattices and Microstructures, Vol. 29, No. 4, 2001

In a semiconductor QD, the confinement-induced exciton binding energy increases abruptly when its geo-metrical sizeR becomes smaller than the bulk exciton Bohr radiusaB [5]. It has been shown by Huet al. [6]that the ground state exciton energyhωoe scaling asR−2 dominates distinctly over the electron–hole (e–h)Coulomb interaction and remains practically independent of the e–h effective mass ratio forR ≤ 0.6aB.Giessenet al. [7] performed a theoretical study of the femtosecond gain spectrum in CdSe QDs under thestrong confinement regime (SCR) withR ≤ 0.5aB and predicted a broad gain spectrum. In the presentanalytical study of pump–probe absorption spectroscopy of the important II–VI and III–V semiconductorQDs, we have selected the QD sizes to be much smaller than the bulk exciton Bohr radii. Such QD sizes areclassified as belonging to the SCR. Under the SCR withR � aB, the QDs are popularly known as smallQDs enabling one to completely ignore the Coulomb interaction [8] and the exciton may be dealt with as acollection of noninteracting e–h pairs [9–11]. At high excitation intensities, the density of the photoinducedexcitons becomes large enough to enhance the probability of formation of quasi-excitonic molecules knownas biexcitons. Wuet al. [12] have shown that the biexciton binding energy also increases sharply under theSCR. Efros and Efros [9] as well as Bányai [8] have shown that under the SCR (i.e.R� aB), the biexcitonground state energyhωob is greater than the exciton ground state energyhωoe, but less than twice as large.Moreover, the biexciton binding energy of small QD may even exceed even the exciton confinement en-ergy [13]. Takagahara [14] presented a detailed analysis of biexciton states in QDs and examined their role inthe enhancement of nonlinear optical properties of the QDs. Using the variational technique, he has also ana-lyzed the dependence of oscillator strength and binding energy on the QD size. Recently, Buttyet al. [15, 16]have reported the pump–probe spectroscopic experimental observation of quasi-continuous optical gain inCdS QDs grown using the sol-gel process in a glass matrix. The gain is observed to be broadened at thelower-energy side of the absorption edge. Herzet al. [17] noted a distinct enhancement of biexciton forma-tion efficiency in QDs under SCR. Currently, intense interest is being shown towards the fabrication of QDarrays with wide application potentiality in the areas of array lasers, array detectors, etc. In such QD arrays,the presence of many QDs of differing size, induces an inhomogeneous broadening. Such broadening canbe incorporated by assuming that the particles have a size distribution around a mean value of the dot radius(R0). Recent experimental observations suggest that a Gaussian distribution of dot sizes fits very well insemiconductor QDs [18].

In Section2, we present the theoretical formulations of the pump–probe spectroscopic technique to ex-amine the nature of absorption/gain spectra in the QD spectrum by using the density matrix approach. Theabsorption spectra in terms of the temporal absorption rate in the QDs in the presence of both excitons andbiexcitons are derived in Section3. Section4 deals with the results and discussion of pump–probe absorptionspectroscopy in real CdS and GaAs QDs. The possible agreement with available experimental observationsis also discussed in the same section. The important conclusions of the present density matrix model arepresented in Section5.

2. Theoretical formulations

The semiclassical density matrix approach has been employed to analyze the pump–probe absorptionspectra of small QDs of direct gap II–VI and III–V semiconductors such as CdS and GaAs. We consider thata strong pump interacts with the QD for timest larger than the population relaxation timeT1 and dephasingtime T2 (t � T1 � T2), such that the equilibrium in the populations of crystal ground and excited e–hpair states is attained. The crystal ground state has been defined in terms of the completely filled valenceband and the empty conduction band. Hereafter, this ground state is referred to as the vacuum state|0〉.The irradiation of a not-too strong pump beam induces a finite population in the excited e–h pair state|e〉.Subsequently, a much weaker broadband probe beam is allowed to shine on the QD with a time delayτ . Thisweak interaction of the delayed probe with the QD can be employed to monitor the optical characteristics ofthe semiconductor QD. A critical analytical study of such phenomena warrants the incorporation of various

Superlattices and Microstructures, Vol. 29, No. 4, 2001 289

important many-body effects such as exciton and biexciton effects. However, the other many-body effectssuch as bandgap renormalization arising due to exchange and correlation processes, phase-space filling orband filling effects become more trivial in QDs than in the bulk crystals. In the present theoretical work,we have investigated analytically the pump–probe absorption/gain spectra in the QDs with finite excitonicand biexcitonic roles while all other many-body effects mentioned above have been neglected without losingmuch quantitative accuracy. It is considered that the photoinduced electronic transitions occur between: (i) thevacuum (|0〉) and excitonic (|e〉) states and (ii) exciton (|e〉) and biexciton (|b〉) states. The formulation onlyconsiders directly allowed photoinduced band-to-band transitions between the crystal ground state and thelowest exciton states with principal quantum numbern = 1 and 2. The role of higher-order discrete excitoniclevels have been overlooked in comparatively narrow bandgap semiconductors exhibiting weakly boundWannier–Mott excitonic structure of the fundamental absorption edge. Moreover, we have ignored the parityviolating direct transitions between the vacuum and the biexcitonic states on the ground that the two-photonabsorption is practically absent. Accordingly, we define the unperturbed Hamiltonian of the QD as

Ho =

( hωo 0 00 hωe 00 0 hωb

)(1)

with hωo(e,b) being the energy of the vacuum (exciton, biexciton) state. The energy of the lowest (1s) excitonsin a QD in the presence of finite e–h Coulomb interaction is obtained as [19]

hωoe= hωg +h2κ2

nl

2mr R2−

1.786e2

εR− 0.248ER. (2a)

Here,hωg andER(= h2/(2mτa2B)) are the crystal bandgap energy and exciton Rydberg energy, respectively.

κnl(= k R; k being the wavenumber) is thenth root of thel th order Bessel function of the first kind withn = 1,2,3, . . . andl = 0,1,2, . . .. In addition,mr = memh/(me + mh) is the effective reduced mass ofthe e–h pair andε = εoεl ; εo andεl being the absolute permittivity and lattice dielectric constant of thesemiconductor. While expressing the 1s exciton energy by eqn (2a), it may be noted thathωoe is the wellknown confinement energy of an e–h pair. On simplification, eqn (2a) can be rewritten as

hωoe= hωg +h2κ2

nl2

2mr R2− (3.572aB/R+ 0.248)ER. (2b)

The energy correction arising due to the e–h Coulomb interaction may be assumed to be negligibly small inthe present formulation on the valid grounds that the model deals with QD spectroscopy under the SCR andthe Coulombic interaction becomes almost trivial forR � aB. Although this approximation ignores someimportant physics of semiconductor QDs, it enables us to obtain the analytical solutions and hence to gatheran important insight into optical absorption/gain spectra [13, pp. 15–16].

For relatively large exciton densities, the Coulombic attraction between two excitons of opposite spins(exciton–exciton interaction) leads to the formation of an exciton molecule popularly known as a biexciton.The biexciton energy is defined as [13]

hωob = 2hωoe−1E, (3)

1E being the biexciton binding energy which under the SCR is positive and larger than the exciton bindingenergyER. In the present calculations, we considered the pump frequency to be in near resonance with thecrystal bandgaphωg and off resonant with the exciton–biexciton transition frequency (ωeb = ωob− ωoe =

ωoe−1E/h) by about(ER +1E)/h. The interaction of the pump field with the QD is represented by thedipole-type interaction Hamiltonian for the three-level scheme as

HI = −

( 0 µoe · E 0µeo · E∗ 0 µeb · E∗

0 µbe · E 0

)(4)

290 Superlattices and Microstructures, Vol. 29, No. 4, 2001

where, EE [= EEo exp(iωt)+c.c.] is the time varying electric field of the pump. The subscripts( j, j ′ = o,e,b)represent the electronic transitions between the states| j 〉 and| j ′〉. The transition dipole moment operator fornear-resonant cases can be defined as|µ j j ′ | = |µ j ′ j | = µ j j ′ with µ j j ′ = ep j j ′/(moω j j ′); p j j ′ andmo beingthe transition momentum matrix element and free-electron rest mass, respectively. The terms representedby µ j j ′ · EEo, are the measures of the QD–pump-field coupling to different exciton and biexciton levels.We assume the fieldEE to be acting parallel to the electric dipole moment operatorµ. In order to study theinteraction of radiation with the three-level QD structure, the 3× 3 density matrix can be defined as

ρ =

(ρoo ρoe ρob

ρeo ρee ρeb

ρbo ρbe ρbb

). (5)

The time-dependent equation of motion of the density matrix is given by [13, pp. 126–132]

ρ = −i

h[H, ρ] −

1

h{Hrel , ρ} = Lρ (6)

with H = Ho + HI ; L is defined as the Louivellian while the curly brackets represent the anticommutationrelation. In eqn (6), we have incorporated the relaxation mechanisms through the Hamiltonian [20]

Hrel =

( hγo 0 00 hγe 00 0 hγb

)with γo, γe andγb denoting the decay constants of the vacuum, exciton and biexciton states, respectively.

The solution of (6) can be obtained by solving the 9× 9 matrix equation. The mathematics involved insolving this matrix equation may be simplified by introducing six new vectors for the three-level schemesimilar to the well known Bloch vectors for two-level pictures and defined as

σ1 = ρoe+ ρeo, σ2 = −i (ρoe− ρeo), σ3 = ρoo+ ρee,

σ4 = ρeb+ ρbe, σ5 = −i (ρeb− ρbe), σ6 = ρee+ ρbb. (7)

The vectorsσ1(4) andσ2(5) have the physical significance of directly yielding the real and imaginary partsof ρoe(eb). Using eqns (6) and (7), the equations of motion of the so-called Bloch vectors have been derivedin the form of a matrix equation given by

σ = Mσ (8a)

with

σ =

σ1σ2σ3σ4σ5σ6

and M =

−γoe 1oe 0 0 0 0−1oe −γoe −2�oe 0 0 0

0 2�oe 0 0 −�eb 00 0 0 −γeb 1eb 00 0 0 −1eb −γeb −2�eb

0 −�oe 0 0 2�eb 0

. (8b)

Here,1oe(eb) = ω−ωoe(eb); �oe(eb) = µoe(eb)Eo/2h is the so-called Rabi frequency in the absence of anydecay or detuning (i.e.1 j j ′ = γ j j ′ = 0) with γ j j ′ = γ j + γ j ′ . In obtaining eqns (8a) and (8b), we haveconfined ourselves to single-photon transitions only. Accordingly, we have ignored the possibility of opticaltransitions among the|0〉 and |b〉 states such as|0〉 → |b〉 and |b〉 → |0〉. Such two-photon biexcitonicabsorption has been studied theoretically at length by Takagahara [14].

The 6× 6 matrix eqns (8a) and (8b) can be solved by employing the Laplace transform technique andassuming a few realistic conditions such as12

oe, �2oe � γ 2

oe and12eb, �

2eb � γ 2

eb. Usage of eqns (7), (8a)and (8b) and extensive mathematical simplifications yield

σ3 =(λ2+ x2

1)(λ2+ x2

2)

λ(λ2+ r 21)(λ

2+ r 22)

(9a)

Superlattices and Microstructures, Vol. 29, No. 4, 2001 291

and

σ6 =2�2

oe(λ+ y1)(λ+ y2)

λ(λ2+ r 21)(λ

2+ r 22); (9b)

λ denoting the Laplace variable. The two solutionsr1 andr2 of eqns (9a) and (9b) are found to be

r 212=β2

oe+ β2eb+ 2(γ 2

oe+ γ2eb)

2±

√[β2

oe+ β2eb+ 2(γ 2

oe+ γ2eb)]

2− 4(β2oeβ

2eb+ 0

4)

2(10)

with +, − signs between the two parts in the right-hand side corresponding tor1 andr2, respectively. Theother terms introduced in eqns (9a), (9b) and (10) are given by

x212=

1

2

[12

oe+ γ2oe+ β

2eb±

√(12

oe+ γ2oe+ β

2eb)

2− 8γ 2oe�

2eb

], (11a)

y12= γeb± i1eb, (11b)

β2oe= 1

2oe+ 4�2

oe+ γ2oe, (11c)

β2eb= 1

2eb+ 4�2

eb+ γ2eb, (11d)

and

04= 4((γ 2

oe+ γ2eb)(�

2oe+�

2eb)−�

2oe�

2eb). (11e)

It is worth mentioning here that the parametersβoe andβeb defined above are analogous to the real part ofthe complex Rabi frequency obtained for a two-level atomic system in the presence of finite detuning anddamping [21]. In the absence of any damping,r1 andr2 reduce to the simple forms as

r 21 ≈ β

2oe≈ (1

2oe+ 4�2

oe) and r 22 ≈ β

2eb≈ (1

2eb+ 4�2

eb) (12)

where1oe(eb) = ω−ωoe(eb). Thus one may infer that the two solutionsr1 andr2 can reveal Rabi oscillationsbetween states|0〉 and|e〉 and between|e〉 and|b〉, respectively, and may be treated as the signatures of theexciton and biexcitons in the semiconductor QDs.

Taking inverse Laplace transforms of eqns (9a) and (9b), we obtain

σ3(t) =1

r 21 − r 2

2

[α1 cosr1t + α2 cosr2t + α3] (13a)

σ6(t) =2�2

oe

r 22 − r 2

1

[β1 cosr1t + β2 cosr2t + β3 sinr1t + β4 sinr2t + β5] (13b)

with

α1 =(x2

1 − r 21)(x

22 − r 2

1)

r 21

, α2 =(x2

1 − r 22)(x

22 − r 2

2)

r 22

,

α3 =x2

1x22(r

21 − r 2

2)

r 21r 2

2

, β1 =y1y2− r 2

1

r 21

,

β2 =y1y2− r 2

2

r 22

, β3 =y1+ y2

r1,

β4 =y1+ y2

r2, β5 =

y1y2

r 22 − r 2

1

. (14)

One may recall from the definitions ofσ3 andσ6 that the measurement of these parameters enables oneself torecognize the active transitions between the states|0〉 and|e〉 as well as that between|e〉 and|b〉, respectively.

292 Superlattices and Microstructures, Vol. 29, No. 4, 2001

Moreover, these two parameters play the key roles in the study of various optical processes such as nonlinearabsorption, emission, etc.

It is apparent from eqns (13a) and (13b) that the population difference will execute superposition of os-cillations at frequenciesr1 andr2. From the definitions in eqn (14), it is evident that the amplitude factorsα1, β1 andβ3 have major contributions from the excitonic states whileα2, β2 andβ4 have their origins inthe biexcitonic states. The above inference is drawn on the basis thatα1, β1 andβ3 are directly dependentuponr1 which is the signature of excitonic characteristics. On the other hand,α2, β2 andβ4 are related to thebiexcitonic parameterr2. It appears worth mentioning that at low excitation intensities, the exciton densityis small, hence the excitonic states play an important role in deciding the optical properties of the mediumwhile at relatively large excitation intensities, the exciton density becomes appreciably high and the exci-ton molecules or biexcitons are formed. This warrants the incorporation of biexcitonic effects even under amoderate (not-too-strong) excitation intensity regime in narrow gap semiconductor nanostructures.

In order to establish the validity of the present formulations, we have reduced the results to that for asimple two-level system and compared the same with the available results [22]. In the absence of biexcitonsand with weak damping mechanisms, the present calculations reduce to those for the ideal single-excitonsystem as may be obtained by using standard Bloch techniques. We find that for single-excitonic systems,the definitions (14) reduce to

α1 = β2oe−1

2oe; α2 = 0; α3 = 1

2oe

β1 = −1; β2 = β3 = β4 = β5 = 0.

Consequently, the two so-called Bloch vectors defined asσ3 andσ6 are found to be

σ3(t) = (1−12oe/β

2oe) cosβoet +1

2oe/βoe2 (15a)

σ6(t) = 2(�oe/βoe)2 cosβoet. (15b)

It is evident from eqns (15a) and (15b) that att = 0, σ3 = 1, implying that the total population is in theground state whileσ6 reduces to zero since�oe = 0 at t = 0. This is consistent with what one may expectfrom the definition ofσ6 as given by eqn (7). It appears appropriate to mention that the analytical treatmentusing the Laplace transformation technique to study pump–probe absorption spectroscopy as presented inthis paper is the first of its kind in terms of its versatility in explaining the optical response in a QD underthe SCR. On this note, we now proceed to calculate the time rate of absorption/gain in small semiconductorQDs irradiated by a strong nearly sharp bandgap resonant pump and a weak broadband probe beam.

3. Pump–probe absorption spectra

The absorption spectra have been obtained by following the standard technique used for a medium irra-diated by an intense (but not-too-strong) driving field. We have considered that this strong pump field offrequencyω shines on the QD for timest � T1, T2 such that a population equilibrium can be achieved.Subsequently, after a time delayτ , a weak probe fieldE′ of frequencyω′ interacts with the QD. The probefield can be used to monitor the absorption characteristics of the small QDs. Accordingly, we have modifiedthe density matrix equation of motion (6) in the presence of the probe fieldE′ as

ρ = Lρ −i

h[H ′I , ρ] (16a)

where

H ′I = −

( 0 µoe · E′ 0µeo · E′∗ 0 µeb · E′∗

0 µbe · E′ 0

). (16b)

Superlattices and Microstructures, Vol. 29, No. 4, 2001 293

Here, we have definedE′ = E′o exp(−iω′T) + c.c. with T = t + τ . The exact solution can be obtained byexpandingρ in various orders as

ρ = ρ(0) + ρ(1) + ρ(2) + · · · (17)

with Lρ(0) = 0. Using eqns (16a) and (17), one can obtain the solution of the first-order density matrix as

ρ(1)(T) = −i

h

∫ T

0eL(T−t ′)

[H ′I (t′+ τ), ρ(0)(t ′)]dt′. (18)

The interaction of the intense pump induces finite polarization in the medium. The net induced polarization inthe presence of the strong pump and weak probe fields under a long-time limit (t � T1, T2) can be obtainedby defining

P(T) =∑n,l

N〈µ〉

=

∑n,l

N limt→∞

tr [µρ(1)(T)] (19)

whereN is the number of dipoles per unit volume. The corresponding time rate of dissipated energy under thepump–probe spectroscopic configuration of the electric fieldsE′(t) characterized by the medium polarizationP(T) is given by [23]

W = E′(t)∂P(t)

∂t. (20)

Using eqns (18)–(20) and mathematical simplifications yield

W = −∑n,l

2Nω′E′2

o

hRe

[∫∞

0[µ2

oeσ3(t′) sin(ω′(t ′ − τ))+ µ2

ebσ6(t′) sin(ω′(t ′ − τ))]dt′

](21)

which under the long-time limit (quasi steady-state) is found to be

W =∑n,l

[2Nω′E′2o

hRe

[1

r 21 − r 2

2

[ω′ cos(ω′τ)

r 21 − ω

′2(µ2

oeα1+ 2µ2eb�

2oeβ1)

+ω′ cos(ω′τ)

r 22 − ω

′2(µ2

oeα2+ 2µ2ebβ2�

2oe)− 2

r1�2oesin(ω′τ)

r 21 − ω

′22µ2

ebβ3

−2r2�

2oesin(ω′τ)

r 22 − ω

′2µ2

ebβ4+cos(ω′τ)

ω′(µ2

oeα3+ 2µ2eb�

2oeβ5)

]]]. (22)

All the relevant terms are already defined in Section2 and τ is the probe delay time. It is evident fromthe above expression that the time rate of absorption will show two resonances atω′ = r1 andω′ = r2,bearing in mind thatr1 andr2 are the signatures of the exciton and biexciton states (as discussed earlier).The summations overn, l are due to the excitonic energy levels forn = 1 and 2;l = 0 and 1 appearing inthe definitions ofhωoe andhωeb. Moreover, the discrete energy level structure of the QD leads to multipleresonances corresponding to different exciton and biexciton states. Hence, the detailed information about thetwo resonances atr1 andr2 can be of interest in the study of exciton and biexciton energy levels of a semi-conductor QD. The time rate of absorptionW as obtained in eqn (22) under the SCR shows its dependenceon the pump–probe delay timeτ . For delay times satisfying the conditionω′τ ≈ 1, the absorption spectrumshould manifest rapid oscillations while for large delay times (ω′τ � 1), small oscillations occurring at afrequencyτ−1 drastically change the absorption characteristics of the QD.

294 Superlattices and Microstructures, Vol. 29, No. 4, 2001

For a better understanding of the absorption phenomena in small QDs under the SCR in the absence ofbiexcitonic effects, we have the simplified form of eqn (22) as

W =∑n,l

2Nω′E′2

o µ2oecos(ω′τ)

hβ2oe

[4�2

oe

β2oe− ω

′2−12

oe

ω′2

]. (23)

One can see from the above expression that the probe frequency (ω′) is in resonance with different values ofRabi frequencyβoe for excitonic levels given byn = 1 and 2; andl = 0 and 1. However, with rising excitonicstates, the magnitude of the factor(µ2

oeE2o/β

2oe) diminishes and hence, the contributions from higher excitonic

levels become negligible. Accordingly, in the next section, while performing numerical calculations, we haveignored the higher excitonic levels withn > 2 andl > 1. This corresponds to the fact that we have restrictedourselves to only the four possible level-to-level transitions, namely, {1,0}, {1,1}, {2,0} and {2,1}.

The theoretical model as discussed above deals only with a single QD. Even though the QD fabricationtechniques have improved to the extent of separating a single QD for single-electron memory devices, widelyused QD devices are based on QD arrays or inhomogeneously distributed QDs [24]. These QD arrays haveattracted immense attention in recent years for the fabrication of fast optoelectronic devices such as QDlaser arrays, detector arrays, etc. In such arrays, the presence of many QDs of different sizes introduces aninhomogeneous distribution of dot sizes which can be accounted for by assuming that the particles have aGaussian size distributionF(R) around a mean sizeR0 with size deviation1R = x R0; x representing thepercentage variation in the Gaussian width. Accordingly, the size-averaged time rate of absorptionW|av fora single QD in an array of many QDs may be defined following the approach of Wuet al. [12] as

W|av =∫ aB

0F(R)W|R d R (24)

whereF(R) = 1R−1√ln 2/π exp[− ln 2 [(R− R0)/1R]2] and known as the probability distribution of QDradii. The present formulations have been developed for a small QDs of dimensionR< aB, and the limit ofintegration in eqn (24) is selected for sizes in the rangeR/aB = 0 to 1.

4. Results and discussions

The numerical analyses of the pump–probe absorption/gain spectra have been made for QD samples ofGaAs and CdS of sizes smaller than the bulk exciton Bohr radii. For the case of CdS QDs, we have takenthe data from the experimental paper of Buttyet al. [16] while for GaAs, standard values from availableliterature have been used and given in Table1. The QD is assumed to be irradiated by a near-resonant laserwhile the probe is tuned from the lowest exciton (withn = 1 andl = 0) to the fourth exciton (n = 2, l = 1)states. The results are calculated for three different cases: (i) assuming the QD to possess a single two-levelstructure (represented by TLS); (ii) an isolated single QD possessing discrete excitonic levels (representedby SQD) and (iii) QDs in an array, in other words, for an inhomogeneously broadened QD (represented byIQD) of finite size distribution width ofx%.

In Fig. 1A and B, we have illustrated the variation of the time rate of absorption with probe wavelength(λ′) for an isolated single QD system by using the complete expression (22) with n = 1 and 2;l = 0 and 1for the QDs of GaAs and CdS, respectively, of sizes smaller than the bulk exciton Bohr radius. For GaAs, itis evident from Fig.1A that two distinct peaks may be observed for the SQD at 805 and 834 nm. These peakscorrespond to the 1s and 1p exciton transitions. One the other hand, a treatment based upon a pure two-levelmodel manifests a single peak at 834 nm. The curve obtained for an ideal two-level system of QDs in CdSshows a single resonance peak at 680 nm while the curve corresponding to SQD manifests two resonancesat 533 and 680 nm, characteristic of the exciton states 2s and 1p, respectively. These features in CdS systemmay be compared with the photoluminescence peaks obtained experimentally by Buttyet al. [16] in CdS

Superlattices and Microstructures, Vol. 29, No. 4, 2001 295

Table 1: Material parameters for QDs of CdS and GaAs used in the numericalcalculations.

Parameter→ hωg aB R0 1E ∼ ER T2 me mhSample↓ (eV) (nm) (nm) (meV) (fs) (×mo) (×mo)

CdS 2.56 2.9 1.7 7.8 72 0.19 0.8GaAs 1.52 12.0 5.6 5.4 200 0.07 0.5

mo—free-electron rest mass

W (

a.u.

)W

(a.

u.)

A

B

Fig. 1.Dependence of average time rate of absorptionW on probe wavelength in A, GaAs and B, CdS QDs.

QDs embedded in a dense glass matrix and in a porous gel. It may, however, be noted that the two peaks inFig. 1B occur in the intermediate regimes of the two bands considered by Buttyet al. [16].

The variation of the average time rate of absorptionW|av with respect to the probe wavelengthλ′ asobtained by using eqns (23) and (24) in both SQD and IQD of GaAs as well as CdS are demonstrated inFig. 2A and B, respectively. The different curves are obtained for dot size distributionx. From Fig.2A, onemy note the presence of an exciton and a biexciton in the SQD of GaAs represented by the two discrete lines

296 Superlattices and Microstructures, Vol. 29, No. 4, 2001

Wav

(a.

u.)

Wav

(a.

u.)

1: SQD2: 5%3: 10%4: 15%5: 20%6: 25%

1: SQD2: 5%3: 10%4: 15%5: 20%

A

B

Fig. 2. Dependence ofWav on probe wavelength in A, GaAs and B, CdS QDs for SQD and IQD having different size distributionwidths.

at 735 and 748 nm, respectively. For the IQD of GaAs, the peaks show red shift that remains almost the samefor 5%≤ x ≤ 25%. However, the peaks are lower as one increases the percentage size distribution. The curveobtained for SQD of CdS in Fig.2B shows a peak at 532.9 nm while the curves obtained for IQD show peakat 552.3 nm, leading to a red shift of 19.4 nm. The physical origin for this red shift in IQD can be understoodfrom the expression for the biexciton energy defined by eqn (3). It may be recalled from the definition ofexciton energy given by eqn (2) that the confinement in terms of QD size increases to the same extent as thatfor smaller QDs, the increase is very large while for low to intermediate size QDs, the correction is almostnegligible and the Coulomb interaction dominates over the confinement. In the present calculations, we haveassumed the average size of the QD to beR0 = 0.6aB and the corresponding pump detuning to the excitonenergy (19.3 nm) matches very well with the experimentally observed red shift of1λ′ = 19.4 nm in CdSQDs, see Buttyet al. [15, 16]. Thus one may infer that the physical mechanism responsible for the red shiftof absorption peaks lies in the QD confinement energy.

Superlattices and Microstructures, Vol. 29, No. 4, 2001 297

∆αd

(a.u

.)∆α

d (a

.u.)

390 kW cm–2

3.5 MW cm–2

660 kW cm–2

16 MW cm–2

A

B

Fig. 3.Absorption/gain spectra of A, GaAs and B, CdS QDs for different pump excitation intensities.

We now turn our attention to the analysis of the absorption/gain spectrum in GaAs and CdS QDs. Theabsorption/gain spectra can be calculated by using the definition of differential transmission spectra (DTS)as [25]

DTS= −1αd = [(W −Wo)/Wo]av (25)

for signal propagation over a distanced. Here,−1α corresponds to the pump induced absorption changeandWo is the time rate of dissipated energy in absence of the pump field and can be calculated from eqn (22)by settingEo = 0. In Fig.3, we have illustrated the features of the pump–probe absorption/gain spectra forsmall QDs of GaAs and CdS in terms of the DTS characterized by the dimensionless change in absorptioncoefficient1αd for a fixed value ofd. It is interesting to note that under the long-wavelength regime, the QDssupport the occurrence of gain. Moreover, the gain behavior is found to be quite sensitive to the change in theexcitation intensity. Both GaAs and CdS QDs support larger gain at higher excitation intensity accompaniedby a blue shift in the probe energy required for the transition from absorption to gain mechanism. The reasonfor the crossover from absorption to gain can be attributed to the definitions ofr1 andr2 as given in eqn (10)as well as that ofW in eqn (22) where the resonance in probe absorption is manifested forω′ = r1 andr2.

298 Superlattices and Microstructures, Vol. 29, No. 4, 2001

Further, from definition (10), it is clear that at large pump intensities, the magnitudes ofr1 andr2 increaseleading to the blue shift in the absorption edge of the probe differential transmission spectrum as displayedin Fig. 3. The increase in gain with intensity may be related to the increasing formation of biexcitons.

5. Conclusions

To conclude, we have made an attempt to theoretically analyze the pump–probe spectroscopic absorp-tion/gain spectra of small QDs of GaAs and CdS duly irradiated by near-resonant strong pump and weakbroadband probe beam. The results are obtained for (i) a two-level system, (ii) a single QD and (iii) an arrayof QDs in the above crystalline samples. The time rate of energy dissipation obtained for a two-level systemshows a single absorption peak while for a single isolated QD, discrete lines are obtained. The incorporationof inhomogeneous broadening as in case of an array of QDs resulted in a broadened absorption/gain spec-tra. The change in absorption obtained using the DTS signal shows a cross over from absorption to gain atthe lower-energy side of the probe spectrum. At large excitation intensities the crossover shows blue shift.The theoretical results of the present work agree quite well with the available experimental results in CdSand CdSe QDs under the SCR. The analytical observations made for GaAs QDs appear quite promising fordevelopment of optoelectronic devices such as optical memories and switches using III–V crystals.

Acknowledgements

The authors are grateful to Dr (Mrs) Pratima Sen for valuable comments on the manuscript. The financialsupport from Department of Atomic Energy (BRNS), Government of India, Mumbai is gratefully acknowl-edged.

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