Absorption spectra of small semiconductor quantum dots

Superlattices and Microstructures, Vol. 26, No. 3, 1999 Article No. spmi.1999.0771 Available online at http://www.idealibrary.com on

Absorption spectra of small semiconductor quantum dots J. T HOMAS A NDREWS†, P RATIMA S EN‡ Department of Applied Physics, Shri G S Institute of Technology & Science, 452003 Indore, India (Received 16 June 1999) A density matrix approach has been employed to study analytically the absorption spectra of small semiconductor quantum dots under the strong confinement regime. The results are obtained for a single quantum dot (SQD) as well as for inhomogeneous distribution of quantum dots (IQDs) with Gaussian distribution of quantum dot sizes. A numerical analysis has been made for a SQD and IQDs in a CdS crystal with data taken from recent experimental work. A negative change in the absorption coefficient occurs in the shorter pump wavelength side of the spectrum due to the biexcitonic contribution. The wavelength at which crossover from positive to negative values of the change in absorption coefficient occurs is found to depend upon both the QD size as well as the excitation intensity. The results agree satisfactorily with the experimental observations in small CdS quantum dots. c 1999 Academic Press

Key words: quantum dots, exciton, biexciton, absorption/gain spectra.

1. Introduction The quantum confinement effects in semiconductors give rise to interesting modifications in electronic and optical properties [1, 2]. In the case of one-dimensional (1D) confinement in semiconductor quantum wells, the ground-state exciton binding energy becomes four times larger than that in the corresponding bulk semiconductor [3]. In a 1D quantum wire, the exciton energy shows a peculiar feature, the energy of the lowest state of the structures becomes negatively infinite showing a divergence of the binding energy [4]. In zero-dimensional (0D) structures, the confinement-induced exciton binding energy abruptly increases for quantum dots (QDs) of sizes smaller than the bulk exciton Bohr radius [5]. In a QD, the discrete levels of electrons and holes localized in all three directions are expected to exhibit distinct peaks in the absorption spectrum. At high excitation intensities, the density of excitons becomes large enough to enhance the biexciton formation probability. Wu et al. [6] have shown that the biexciton binding energy increases abruptly in the strong confinement regime where the size of the QD represented as R is smaller than the bulk exciton Bohr radius aB . While studying small quantum dots in the strong confinement regime, Banyai [7] showed that the biexciton energy will be greater than the exciton energy but less than twice the energy of the exciton. The biexciton binding energy of a small QD is also found to exceed the confinement energy. Recently, Butty et al. [8, 9] have reported a quasicontinuous optical gain in CdS quantum dots fabricated by the sol–gel process and embedded in a glass matrix. The gain is observed to broaden at the lower energy side of the † Present address: Department of Applied Physics, Birla Institute of Technology, Mesra, Ranchi 825215, India. ‡ E-mail: pratima [email protected]

0749–6036/99/090171 + 10 $30.00/0

c 1999 Academic Press

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absorption edge. Herz et al. [10] also observed a distinct enhancement of biexciton formation efficiency in smaller QDs with R  aB as compared to that in a large QD with R  aB . A realistic QD system contains quantum dots of various radii making it necessary to incorporate a quantum dot size distribution function. In such quantum dot arrays, the presence of many QDs in the surrounding induces an additional broadening. This can be accounted for by assuming that the particles have a size distribution around a mean value of dot radius. The above discussions reveal the important role of the biexcitonic effect and hence warrants its proper incorporation when considering the optical properties of small QDs. The density matrix approach has been employed to analyze theoretically the spectra of a small semiconductor quantum dot. The numerical analyses made for both a single quantum dot (SQD) and an inhomogeneous distribution of quantum dots (IQDs) reveal that the results are in good qualitative agreement with the recent experimental observations in a CdS QD system [8, 9].

2. Theoretical formulations The semiclassical density matrix approach has been employed to study the role of biexciton on the radiation–matter interaction in small semiconductor quantum dots. We consider the optical transitions to occur between the crystal ground state and a single electron–hole pair state (i.e. excitonic state) and that between exciton and biexciton states. Accordingly, a 3 × 3 matrix has been introduced for the transition dipole moment (µ) ˆ and density matrix operator (ρ). The expressions for the induced polarization in a single QD as well as in inhomogeneous distribution of QDs have been obtained. Recent experiments with semiconductor QDs show an increased contribution of the biexciton transition to the photoluminescence line shape as compared to the bulk and quantum well semiconductors [10]. For a small QD of radius R much smaller than the bulk exciton Bohr radius aB , the exciton energy ~ωoe can be defined as [11–13]:   κnl aB 2 ~ωoe = ~ωg + E R , (1) R where ~ωg is the band gap energy, E R [= ~2 /(2m r aB2 )] is the exciton Rydberg energy and m r is the reduced effective mass. κnl is the nth root of the lth order Bessel function with n and l corresponding to the 1s, 1p, 1d, . . . , 2s, 2p, 2d, . . . levels of the electrons and holes [11–13]. The biexciton energy (~ωob ) is defined as ~ωob = 2~ωoe − 1E,

(2)

with 1E being the biexciton binding energy, which is a measure of the inter-band Coulomb effect. From (1) one can argue that for a SQD, the energy shift due to the confinement energy will be much larger than the biexciton binding energy which is related to the Coulomb energy. The lower and upper limits for the biexciton molecular binding energy 1E in a small quantum dot was proposed as [7, 12]:     me + mh me + mh 2+ E R Cl () < 1E < 2 + E R Cu (). (3) 2m r 2m r Here,  is the dielectric constant. Numerical evaluations for  = 10 show that Cl (10) = 0.104 and Cu (10) = 0.71 [5]. For the electron–hole effective mass ratio m e /m h = 0.1, the lowest molecular binding energy becomes comparable to the bulk exciton binding energy. Hence, in the strong confinement regime one cannot ignore the biexciton Coulomb energy [5]. In the present formulations, we have assumed that the QD possesses discrete energy levels which scale with the inverse square of the radius. The different excitonic levels arise as a consequence of the spherical confinement potential. The annihilation of a photon leads to the creation of an electron–hole pair known as an exciton. In the presence of a large number of excitons, the Coulomb energy existing between two exciton pairs unite them to form an exciton molecule (which is well known as a biexciton molecule).

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Incorporating the exciton and biexciton states of a QD, we have defined the semiclassical Hamiltonian as H = Ho + H I ,

(4)

with H0 =

~ωoo 0 0

0 ~ωoe 0

0 0 ~ωob

!

,

(5)

and ! 0 −µoe E 0 ∗ ∗ H I = −µeo E . (6) 0 −µeb E 0 −µbe E 0 Here, ~ωoo , ~ωoe and ~ωob are the unperturbed energies of the ground, exciton and biexciton states, respectively. µoe and µeb are the transition dipole moments for the ground to exciton and exciton to biexciton transitions. E is the amplitude of the electromagnetic field and is assumed to act parallel to the transition dipole moment operators. In the present formulations, we have considered the interaction of a single photon with the semiconductor QD which leads to the transition of an electron from the ground state |0i to the exciton state |ei. The lifetimes of these states are taken as γe−1 and γb−1 . The transitions from ground to biexciton state and biexciton to ground state are not directly allowed optical transitions and hence have been neglected. Accounting for the relaxation mechanisms, the density matrix equation of motion can be written as 1 i ρ˙ = − [H, ρ] − {Hr el , ρ}, ~ ~

(7)

with

! ! ρoo ρoe ρob γo 0 0 ρ = ρeo ρee ρeb , and Hr el = 0 γe 0 , (8) ρbo ρbe ρbb 0 0 γb where the diagonal matrix Hr el has been incorporated phenomenologically to account for various relaxation processes [14]. Usage of (4)–(8) yields the equation of motion as a 9 × 9 matrix equation ρ˙ = (i M − Mr el )ρ, ˜

(9a)

with 0 −oe 0 oe 0  −oe −1oe −eb  −eb −1ob 0  0  0 0 1oe  oe  M = 0 oe 0 −oe  0 oe 0  0  0 0 eb  0  0 0 0 0 0 0 0 0  2γo 0 0 0 0 0 0 0 0  0 γoe 0  0 γob 0 0 0  0  0 0 γoe 0 0  0  Mr el =  0 0 0 0 2γe 0  0 0 0 0 γeb  0  0 0 0 0 0  0  0 0 0 0 0 0 0 0 0 0 0 0 

0 0 0 oe 0 0 0 oe 0 −oe 0 eb 0 −eb 0 −eb −1eb 0 0 0 1ob eb 0 −oe 0 eb 0  0 0 0 0 0 0   0 0 0   0 0 0   0 0 0 ,  0 0 0   γob 0 0   0 γeb 0 0 0 2γb

0 0 0 0 eb 0 −oe 1eb −eb

and

 0 0   0   0   0 ,  eb   0   −eb 0   ρoo  ρoe     ρob     ρeo    ρ˜ =  ρee  .    ρeb     ρbo    ρbe ρbb

(9b)

(9c)

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Here, 1oe = ω − ωoe , 1eb = ω − ωeb , ωeb = ωob − ωoe , i j = µi j E/~, ω is the pump frequency and γi j = γi + γ j . In obtaining eqns (9), we have assumed that |µi j | = |µ ji | and |µob | = |µbo | = 0. Knowledge of the terms such as ρoe and ρeb can be suitably exploited in obtaining the induced polarization and consequently, one can study the absorption spectra of QDs. On the other hand, the diagonal elements ρoo , ρee and ρbb can be utilized to describe the emission spectra of the signal. The equations of motion of the density matrix obtained as expressed by eqns (9a)–(9c) become mathematically involved. Such an equation with 9 × 9 matrix elements can be solved under suitable approximations [15]. The relaxation terms γo , γe and γb considered here have their origin in the finite lifetime characteristic of the corresponding state. In the absence of the biexcitonic effect and for a single excitonic state, the present formulations reduce very well to the standard Bloch equations obtained for a two-level system [16]. By comparing eqns (1.14) of Brewer [16] and eqns (9) of the present calculations, we find that (o) (o) [(ρoo − ρee ) − (ρoo − ρee )]/T1 = 2(γo ρoo − γe ρee ) and T2−1 = γo + γe . In general, T1 and T2 are defined as the recombination lifetime and the dephasing time, respectively. For semiconductors, T1 is of the order of nanoseconds while T2 is found to be in the pico-femtoseconds regime [17]. Thus, one can note that the dominating relaxation term corresponds to T2−1 only. Accordingly, we have retained the dephasing decay rates in the equations of motion of the density matrix. The mathematically involved equations of motion (9) can be solved by introducing the real quantities R1 , R2 , . . . , R6 very similar to those introduced in the vector model of density matrix equations [16, 18] defined as R1 = ρoe + ρeo ,

(10a)

R2 = −i(ρoe − ρeo ), R3 = ρoo − ρee .

(10b) (10c)

The vectors R1 , R2 and R3 are exactly the same Bloch vectors as used for the study of the interaction of radiation with a two-level atomic system. In the present model, we are dealing with a three-level structure where the photoinduced electron–hole transitions occur between the ground to single electron–hole pair states (known as excitonic states) as well as between the excitonic state to the two-electron–hole pair states (known as biexcitonic states). In order to account for the later transitions, we have introduced three new vectors R4 , R5 and R6 , similar to the Bloch vectors for the atomic system as R4 = ρeb + ρbe , R5 = −i(ρeb − ρbe ), R6 = ρee − ρbb .

(11a) (11b) (11c)

Employing the above definition of R, eqns (9) may be expressed by  ˙   R1 −γoe −1oe 0 0 0 γoe −2oe 0 0  R˙ 2   −1oe  ˙   −2oe 0 0 −eb  R3   0  ˙ = 0 0 −γeb −1eb  R4   0  ˙   R5 0 0 0 −1eb γeb R˙ 6 0 −oe 0 0 −2eb

  0 R1 0   R2    0   R3   . 0   R4    −2eb R5 0 R6

(12)

In obtaining (12), we have ignored the density matrix components ρob and ρbo assuming that such transitions are forbidden in the present context. The parameters R1 and R4 have the physical significance of directly yielding the real part of ρoe and ρeb while R2 and R5 yield the imaginary parts. Taking Laplace transform on both the sides of (12), we obtain Rˆ 1 = 21oe oe

2 s 2 − 42eb − 12eb − γeb

s(s 2 − r12 )(s 2 − r22 )

,

(13a)

Superlattices and Microstructures, Vol. 26, No. 3, 1999 2 s 2 − 42eb − 12eb − γeb

Rˆ 2 = −2oe

Rˆ 4 =

175

(s 2 − r12 )(s 2 − r22 )

,

42oe 1eb eb , s(s 2 − r12 )(s 2 − r22 )

(13b)

(14a)

and 42oe eb , (14b) (s 2 − r12 )(s 2 − r22 ) where the upper carets indicate the Laplace transformed variable and s is the Laplace variable. The two roots r1,2 in eqns (14) are defined as r q 1 2 β2 , r1,2 = √ α 2 ± α 4 − 4βoe (15) eb 2 with Rˆ 5 =

2 2 α 2 = βoe + βeb , 2 βoe 2 βeb

= =

12oe 12eb

+ 42oe + 42eb

(16a) 2 + γoe , 2 + γeb .

(16b) (16c)

Substitution of (16) in (15) simplifies the definition of r1,2 as r1 = βoe

and

r2 = βeb .

(17)

Equations (16) and (17) reveal distinctly that the two roots r1 and r2 have their origins in the excitonic and biexcitonic states, respectively. Taking the inverse Laplace transforms of (13)–(14), we find   2 2βeb 2oe 1oe R1 = cos βoe t + cos βeb t − 1 , (18a) 2 2 − β2 βoe βoe eb oe R2 = −2 sin βoe t, (18b) βoe   2 2 βeb 42oe eb 1eb βoe R4 = cos βoe t − cos βeb t + 1 , 2 β2 2 − β2 2 − β2 βoe βoe βoe eb eb eb

(19a)

and 42oe eb 42oe eb sin β t + sin βeb t. (19b) oe 2 − β2 ) 2 − β2 ) βoe (βoe βeb (βoe eb eb The components of the R-vector represented by eqns (18) and (19) oscillate at the frequencies βoe and βeb . Following standard practice [5], the total polarization has been obtained from the ensemble average of the transition dipole moment by defining " # X X P= T r (ρµ) = (µoe ρoe + µeb ρeb + c · c). (20) R5 =

e,b

e,b

Using (18)–(19) in (20), we obtain P(= o χ ε) =

X e,b

(Po + P1 + i P2 ),

(21a)

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with 4µeb 1eb eb 2oe 2µoe 1oe oe + , 2 2 β2 βoe βoe eb   2µoe 1oe oe 4µeb 1eb eb 2oe P1 = − cos βoe t, 2 2 (β 2 − β 2 ) βoe βeb oe eb   4µoe 1oe oe 2eb 4µeb 1eb eb 2oe + − cos βeb t, 2 (β 2 − β 2 ) 2 (β 2 − β 2 ) βoe βoe oe oe eb eb   4µeb 2oe eb 2µoe oe 4µeb 2oe eb − sin β t + sin βeb t. P2 = − oe 2 − β2 ) 2 − β2 ) βoe βoe (βoe βeb (βoe eb eb

Po = −

(21b) (21c)

(21d)

The net induced polarization P contains the contributions from both excitonic as well as biexcitonic states. The induced polarization as expressed in (21) comprises of three distinct components. The first two components are real and hence responsible for the linear as well as the nonlinear refractive indices of the medium while the third term (P2 ) is imaginary and characterizes the nature of the absorption process. One can further note that (P2 ) contains linear as well as nonlinear terms. The nonlinear terms contain the biexcitonic transition dipole moment and can either enhance or reduce the net absorption. The change in absorption 1α arising due to nonlinearity can be calculated by deducting the linear absorption from the total absorption obtainable from P2 . In order to check the validity of the present formulations, we neglect the biexcitonic contribution in (21) by assuming 1eb = eb = γb = 0, and find that   X 21oe oe oe P0 = µoe [cos β t − 1] − 2i sin β t (22) oe oe . 2 βoe βoe e For a particular value of e, the above expression matches well with the standard expression for the transition probability in a two-level atomic system [16, 18]. In obtaining (21), we have assumed that a single mode of the electromagnetic field interacts with a single quantum dot. Thus, the absorption spectra obtainable by using (21) is expected to yield a discrete spectrum. The available nanostructure fabrication technique gives very little scope to isolate a single QD. Therefore, one usually prefers to work with an inhomogeneous distribution of quantum dots for the experimental studies. In such cases, the presence of many QDs surrounding every individual QD leads to an additional inhomogeneous broadening. In order to account for collective effects arising from such IQDs, one has to incorporate a size distribution factor F(R) with R being the QD size. The inhomogeneous broadening may be taken into account by assuming that the particles have a Gaussian size distribution F(R) around a mean value 1R = x Ro where x is the percentage (%) variation in the Gaussian width and Ro is the average dot size [12]. Accordingly, we define the complex susceptibility χ obtainable from the induced polarization P in the strong confinement regime as Z aB χs = χ| R F(R)d R, (23) 0 √ with F(R) = 1R −1 ln 2/π exp[− ln 2((R − Ro )/1R)2 ]. In the steady-state regime, when the irradiation time t p is much larger than the population relaxation times (t p  γa−1 , γb−1 ), the oscillating terms can be averaged out. Using (21) and (23), we can study the absorption spectra and the transmitted intensity from a single QD as well as from an inhomogeneous distribution of QDs. In sol–gel-derived Cds QDs, Butty et al. [8] reported the experimental observation of the bleaching of absorption spectrum around 465 nm. They also reported the optical gain at the low energy side and explained it on the basis of the biexcitonic gain. We have applied our analysis to a QD of CdS and to make a qualitative comparison of the experimental observation by Butty et al. [8, 9] with those obtained in the present theoretical

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analysis, we have taken the values of the following parameters as those used by Butty et al. [8, 9]: crystal band gap energy ~ωg = 2.56 eV, aB = 2.9 nm, quantum dot radius R = 1.75 nm, inhomogeneous QD size distribution width in the range of 30% to 50% and excitation intensity in the range of 190 kW cm−2 to 30 MW cm−2 . The other material parameters needed for the numerical estimations are m e = 0.235 m o , m h = 1.35 m o [19] and γe−1 = 2.27 ps [5]. The biexciton lifetime is assumed to be 0.1 γe−1 . The different roots of the Bessel function are: κ10 = π, κ11 = 4.4934, κ12 = 5.7635, κ20 = 6.2832, κ21 = 7.7253, κ22 = 9.0950, κ30 = 9.4248, etc. The variation of |χ |2 as a function of wavelength has been plotted in Fig. 1 for a single (SQD) and inhomogeneous distribution of the QDs (IQDs) at different size distribution widths. In obtaining these curves, we have chosen n = 1, 2 and l = 0, 1 accounting for four excitonic and four biexcitonic levels. The energy of excitonic levels is calculated by using (1) while that of the biexcitonic level is approximated from (2) and (3), considering that the lattice dielectric constant  ≈ 10 [8]. The optical transitions can take place between the crystal ground state and excitonic states as well as between the excitonic and biexcitonic states. The wavelengths corresponding to these transitions (λoe and λeb ) are calculated as λoe = 391, 325, 247 and 198 nm, and λeb = 395, 328, 249 and 199 nm, respectively, for a single QD with R = 0.603aB . It is worth mentioning here that λoe and λeb are highly sensitive to the quantum dot size. The peaks around 391 nm and 326 nm in Fig. 1 demonstrate the existence of exciton and biexciton states. It may be easily noted from the figure that the absorption peak at the shorter wavelength side in the case of a SQD almost loses its identity when one replaces it by an IQD with a large QD size distribution width. It is worth mentioning that the average size of the IQDs is the same as that of the SQD. However, a Gaussian type of distribution can be noted for the IQDs in the longer wavelength regime, where the SQD peak is also very prominent. In order to study the absorption spectra for CdS QDs analytically, we have calculated the change in the absorption coefficient 1α by subtracting the linear absorption αo (ω) from the total absorption coefficient α(ω) as may be derived from (21) and (23). The variation of 1α as a function of wavelength has been plotted in Fig. 2 at two different pump excitation intensities, 190 kW cm−2 and 30 MW cm−2 . From Fig. 2, it may be observed that 1α becomes negative at a certain pump wavelength and after attaining a maximum negative value it tends to saturate at the zero value at a relatively larger pump wavelength. Moreover, an increase in pump intensity leads to a blue shift in the crossover value of the wavelength with a reduced peak value of 1α. A critical examination of (21d) reveals that 1α is strongly dependent upon the exciton–biexciton (e–b) transitions. Therefore, increased exciton generation at higher pump intensities enables stronger e–b transition leading to the larger negative value of 1α at a given pump frequency. Quite interestingly, such a feature in the absorption characteristics of a CdS quantum dot has been observed experimentally (shown in the inset) by Butty et al. [8, 9], except that their absorption/gain spectra did not reveal the occurrence of any peak in the negative value of the absorption coefficient. It should be noted that the theoretically obtained crossover wavelengths are shorter than the experimental observations. It may be recalled in this connection that the experimental spectra were observed using a pump and probe geometry and the absorption/gain spectra were deduced by subtracting the differential transmission spectrum (DTS) signal from the linear absorption spectrum [8, 9]. The absorption spectra for different dot sizes, viz., R = 0.50aB , 0.55aB and 0.60aB at an excitation intensity of 30 MW cm−2 are exhibited in Fig. 3. This figure also confirms the negative values of 1α in the shorter wavelength side of the spectrum. The crossover from positive to negative values of 1α shifts to higher wavelength with decreasing dot size. To conclude, the effect of biexcitonic states on the absorption spectra of semiconductor quantum dots under the strong confinement regime has been analytically investigated and the following inferences may be drawn.

Superlattices and Microstructures, Vol. 26, No. 3, 1999

SQD IQD 30% IQD 40% IQD 50%

1.0 0.8 0.6

326

|χ|2 (a.u.)

391

178

0.4 0.2 0.0 400

350

300

450

λ (nm)

Absorption α/Gain –g (cm–1)

Fig. 1. Pump wavelength dependence of |χ|2 for a single quantum dot of CdS (solid) as well as for an inhomegeneous distribution of quantum dots (dashed) at an excitation intensity of 30 MW cm−2 .

200

1α (cm–1)

100

250 200 150 100 no pump with pump

50 0 –50 400

450 500 Wavelength (nm)

550

0

–100

190 kW cm–2 30 MW cm–2

–200 400

420

440

460

480

500

λ (nm) Fig. 2. Change in absorption coefficient 1α with pump wavelength for an inhomegeneous distribution of quantum dots of CdS. The inset shows the experimental data of Butty et al. [9]. The average size of the QD is Ro = 0.60aB , with dot size distribution 40%. The curves are obtained for different excitation intensities.

(i) The value of |χ |2 obtained for inhomogeneous distributions of the quantum dots demonstrate broadening with its origin being in the finite size distributions. (ii) A negative change in the absorptive behavior occurs at a shorter pump wavelength due to the finite biexcitonic contribution. (iii) The wavelength at which crossover from positive to negative values of 1α takes place is found to depend upon both the QD size as well as the excitation intensity. (iv) The theoretical analysis has been found to exhibit qualitative agreement with the recent experimental observations of the optical absorption/gain characteristics in small CdS quantum dots.

Superlattices and Microstructures, Vol. 26, No. 3, 1999

179

200

0.50 aB 0.55 aB

1α (cm–1)

100

0.60 aB

0

–100

–200 390

420

450

480

λ (nm) Fig. 3. Wavelength dependence of the change in absorption coefficient 1α in an inhomogeneous distribution of QDs, with 40% dot size distribution at an excitation intensity of 30 MW cm−2 . The curves are obtained for different dot sizes of the quantum dot.

Acknowledgements—The authors are very grateful to Professor Pranay K. Sen for fruitful discussions and a critical reading of the manuscript. The financial support from University Grants Commission (PS), Council of Scientific and Industrial Research, New Delhi and Department of Atomic Energy, Mumbai are gratefully acknowledged.

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