TIME-RESOLVED TERAHERTZ SPECTROSCOPY OF
SEMICONDUCTOR QUANTUM DOTS
by
GEORGI DAKOVSKI
Submitted in partial fulfillment of the requirements
For the degree of Doctor of Philosophy
Thesis advisor: Dr. Jie Shan
Department of Physics
Case Western Reserve University
January, 2008
CASE WESTERN RESERVE UNIVERSITY
SCHOOL OF GRADUATE STUDIES
We hereby approve the dissertation of
______
candidate for the Ph.D. degree *.
(signed)______(chair of the committee)
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(date) ______
*We also certify that written approval has been obtained for any proprietary material contained therein.
Table of Contents
List of figures……………………………………………………………………5
Abstract………………………………………………………………………..10
1. Introduction to optical pump/terahertz probe spectroscopy………………12
1.1 Generation and detection of THz radiation……………………….13
1.2 Applications of THz-TDS…………………………………………...19
1.3 Outline of the thesis………………………………………………...20
2. Localized THz generation via optical rectification in ZnTe………………..22
2.1 Difference-frequency generation…………………………………..24
2.2 Experimental setup………………………………………………….26
2.3 Results………………………………………………………………..27
2.4 Numerical simulation and discussion……………………………..30
2.5 Conclusions………………………………………………………….39
1 3. Finite beam-size effects in optical pump/THz probe
spectroscopy………………………………………………………………….41
3.1 Theoretical model…………………………………………………...42
3.2 Experimental setup………………………………………………….45
3.3 Results and discussion……………………………………………..46
3.4 Conclusions………………………………………………………….49
4. Introduction to semiconductor quantum dots……………………………...51
4.1 Fabrication and Characterization………………………………….53
4.2 Applications………………………………………………………….57
4.3 Electronic structure of semiconductor quantum dots…………....59
4.3.1 Parabolic band model……………………………………..61
4.3.2 Multiband model for CdSe QDs…………………………..66
4.3.3 Multiband model for PbSe QDs…………………………..71
4.4 Limitations on the use of the kp⋅ method…………………..…..76
5. Terahertz electric polarizability of excitons in CdSe and PbSe
quantum dots………………………………………………………………….78
5.1 Response of a single exciton to THz radiation…………………...80
5.2 Experimental setup………………………………………………….81
5.3 Results and discussion……………………………………………..82
2 5.4.1 Calculations based on the parabolic band model……....88
5.4.2 Calculations based on the multiband models.……….....90
5.5 Conclusions………………………………………………………….92
6. Multiexciton Auger recombination in semiconductor
quantum dots studied with THz-TDS……………………………………….94
6.1 Experimental setup………………………………………………….96
6.2 Results…………………..……………………………………………96
6.3 Discussion……………………………………………………….....100
6.4 Conclusions………………………………………………………...103
7. Response of multiple excitons in CdSe nanoparticles
studied with terahertz time-domain spectroscopy……………………….104
7.1 Experimental setup………………………………………………..105
7.2 Results………………………………………………………………105
7.3 Data analysis……………………………………………………….108
7.4 Inclusion of the Coulomb interaction and discussion…………..109
7.5 Conclusions………………………………………………………...113
3 8. Size-dependence of degenerate two-photon absorption in
semiconductor QDs…………………………………………………………114
8.1 TPA in bulk materials………………………………………………116
8.2 Optical pump/white-light probe technique……………………….117
8.3 Experimental setup and details…………………………………..118
8.4 Results…………………………………………………………...... 119
8.5 Model for TPA in QDs……………………………………………...125
8.6 Conclusions………………………………………………………...129
9. Summary……………………………………………………………………..130
References…………………………………………………………………………....133
4 List of Figures
1. Fig. 1.1 Standard THz-TDS spectrometer, based on an amplified laser system.
THz radiation is generated through optical rectification and detected through
electrooptic sampling in ZnTe.
2. Fig. 2.2.1 Sketch of the THz emitter employed in the measurement (not to
scale): optical pump beam (800 nm) is focused onto a 20 μm ZnTe emitter
( (110) orientation) on a 500 μm ZnTe substrate ( (100) orientation).
Translation of the lens allows variation of the effective size of the emitter.
3. Fig. 2.3.1 Normalized spectral density E( ω ) 2 versus the FWHM of the
optical excitation a . Three representative frequencies are shown in
symbols. Solid line is a fit to Eq. (2.4.16). Inset: size dependence of the
total THz power (normalized) in the regime of two-photon absorption using
100% (filled) and 50% (open) of the available excitation power.
4. Fig. 2.3.2 Characteristic size of the emitter ac as defined in the text for the
transition of spectral density behavior (from a0 to a−2 dependence) versus
the THz wavelength λTHz . Symbols: experiment; line: simulation.
5. Fig. 2.4.1 Typical layered geometry utilized in our experiment.
6. Fig. 2.4.2 (a): Simulated spectral density E( ω ) 2 at 0.9, 1.4, and 1.9 THz
(solid lines) versus the FWHM of the optical excitation a for a 30o collection
5 angle. (b): simulated spectral density E( ω ) 2 at 1.4 THz for collection
angles of 30°, 15°, and 7.5°. Dashed lines illustrate a−2 dependence.
7. Fig. 3.1.1 Simulation of the radiation power at 0.45 (dashed line) and 1.15
THz (solid line) emitted from a source of a fixed peak polarization but varying
sizes. The radiation is collected by off-axis parabolic mirrors of 64 mm in
diameter and 120 mm in focal length as used in the experiment. Critical
4 2 sizes ac for the transition from the a to the a dependence are indicated for
both frequencies.
8. Fig. 3.3.1 (a) Complex conductivity of photo-induced carriers in CdTe at room
temperature as a function of frequency for pump sizes of 3.6mm FWHM (solid
lines) and 0.7mm (dashed lines). Fit to the Drude model is shown for 3.6mm
pump (grey lines). (b) Pump size dependence of σ( ω ) 2 at 0.45 (empty
circles) and 1.15 THz (filled circle). Lines: fits to Eq. (3.1.1).
9. Fig. 3.3.2 Concentration (empty circles) (a) and mobility (empty squares) (b)
of photo-induced carriers in CdTe inferred from the fit of the measurement to
the Drude model over a range of pump beam sizes. Data that cannot be
described by the Drude model are indicated by a ‘x’; line (a): fit of the
extracted density to the model as described in the text; (filled circles) (a): peak
carrier density inferred from our model.
10. Fig. 4.1 Linear absorption of colloidal CdSe/ZnS nanoparticles of varying
sizes dissolved in hexane.
6 11. Fig. 4.1.1 a) High resolution TEM image of PbSe QDs, revealing the lattice
structure; b) Low resolution image, showing an ensemble of QDs [88].
12. Fig. 5.3.1 THz electric-field waveform transmitted through an unexcited
suspension of 3.1-nm-radius PbSe QDs (black solid line) and the
photoinduced change in this waveform (grey solid line). Inset: dependence
of the pump-induced response on the pump-probe delay time in the first 10 ps.
(b) Spectral dependence of the photoinduced complex dielectric response of
the QD suspension (solid grey lines). Dashed lines in (b) represent a model
with Δε'const= and Δε'' = 0 and the dashed line in (a) is the change in the
THz electric-field waveform predicted by the model.
13. Fig. 5.3.2 Schematics of the electric polarizability of an exciton strongly
quantum confined in a QD of radius R to an externally applied electric field
F . Within the parabolic-band effective mass model as described in the text,
in the absence of the field (left), the ground-state exciton possesses a
spherically-symmetric electron and hole distribution centered at the center of
the dot (only the electron distribution shown) which result in a zero net dipole
moment. The externally applied electric field perturbs the exciton wave
function and the charge spatial distribution (right), which leads to a net dipole
moment P in the QD.
14. Fig. 5.3.3 Polarizability of photoinduced excitons strongly quantum-confined
in CdSe and PbSe QDs as a function of the QD radius. Details about the
7 experimental results (symbols) and the theoretical results based on a
parabolic-band effective mass model (dotted lines) and multiband effective
mass models (solid lines) are described in the text.
15. Fig. 6.2.1 THz electric-field waveform transmitted through an unexcited
suspension and the photoinduced change in the waveform (red line) for an
average of <>=N.15 excitons per QD. Inset: spectral dependence of the
photo-induced complex dielectric response of the QD suspension (solid lines).
Dashed lines in the inset represent a model with Δε'const= and Δε'' = 0 ;
dashed line in the main panel is the change in the THz electric-field waveform
predicted by the model.
16. Fig. 6.2.2 Pump-dependence of different average exciton populations in a
3.3-nm CdSe QD.
17. Fig. 6.2.3 Dependence of the normalized real part of the induced complex
dielectric response (by the molar concentration) on the average number of
excitons per QD (black dots). Red line: fit to a power law, y = Ax p , p ≈ 05. .
18. Fig. 6.3.1 Size-dependence of multiexciton recombination lifetimes obtained
via THz-TDS (red symbols). Black symbols: data obtained via transient
absorption spectroscopy. Straight lines: fits to a R3 dependence.
19. Fig. 7.2.1 Dependence of the normalized real part of the induced complex
dielectric response (by the molar concentration) on the average number of
excitons per QD (red dots). Black dots: recalculated exciton population,
8 accounting for the Auger recombination. Solid line: model of non-interacting
carriers.
20. Fig. 7.4.1 Dependence of the hole polarizability on the number of carriers
calculated with the parabolic-band and the multiband effective mass models
for a 2.58-nm QD. In both models the conduction band is treated with the
help of the parabolic-band model.
21. Fig. 8.4.1 Transient absorption dynamics at the first exciton transition of
3.3-nm-radius CdSe QDs induced by 400-nm (solid line) and 800-nm (dashed
line) excitation. Inset: linear absorption spectrum of the QD suspension;
arrows identify the pump and probe wavelengths.
22. Fig. 8.4.2 Pump fluence dependence of the normalized absorption −Δα / α0
at the first exciton transition of 2.7-nm-radius CdSe QDs induced by 400-nm
(squares) and 800-nm (triangles) excitation. Solid and dashed lines are
linear and quadratic fits.
23. Fig. 8.4.3 (a) Dependence of the two-photon absorption cross section σ()2 at
800 nm on the QD radius; (b) Dependence of the two-photon absorption
coefficient β at 800 nm on the QD radius. Circles: experiment; solid line:
parabolic-band effective mass model as described in the text; triangles: β of
bulk CdSe at 800 nm.
9 Time-Resolved Terahertz Spectroscopy of
Semiconductor Quantum Dots
Abstract
by
GEORGI DAKOVSKI
Spectroscopy in the far-infrared part of the electromagnetic spectrum based on the time-domain measurements of transient terahertz pulses has become a standard experimental technique. In the first part of this thesis we present results regarding applications of this technique to the problem of near-field, sub-wavelength imaging and the effect of finite-size beams in optical pump/terahertz probe experiments. The second part presents time-resolved far-infrared measurements performed on semiconductor quantum dots.
Amongst many applications, terahertz time-domain spectroscopy (THz-TDS) has been successfully used for imaging. We present a method based on highly-localized THz generation through a nonlinear process that achieves sub-wavelength resolution and a favorable power throughput, essential for the sensitivity of the measurement. With respect to standard optical pump/THz probe measurements the finite size of the beams intersecting at the sample introduces non-trivial effects. We modeled the problem as THz-induced
10 radiation from the optically-generated polarization to obtain useful requirements for the relative dimensions of the pump and probe beams that allow readily interpretable measurements.
The ability to directly measure the electric field of the THz pulse opens the possibility to perform spectroscopic measurements with picosecond time-resolution. We used this feature of the THz-TDS to explore the response of short-lived, optically-induced excitons to external THz electric fields in colloidal semiconductor quantum dots (QDs). In the limit of single exciton per QD we performed a comparative study between two systems of QDs, CdSe and PbSe, possessing different electronic structure. For both samples the response was found to be atom-like, and was successfully simulated with the help of an effective-mass model. The presence of multiple excitons within a QD is accompanied by strong many-body interactions manifested by the extremely fast
Auger recombination. We investigated the characteristic depopulation rates in
CdSe QDs by the THz-TDS and compared to existing data obtained via other techniques. This allowed us to explore the response of multiple excitons within a
QD and evaluate the importance of the carrier-carrier Coulomb interaction.
Finally we employed the optical pump/white light probe method to study the size-dependence of the two-photon absorption coefficient in CdSe QDs, and compared experiment to a parabolic-band model.
11 Chapter 1
Introduction to Time-Resolved Optical Pump/Terahertz Probe
Spectroscopy
Terahertz time-domain spectroscopy (THz-TDS) is a spectroscopic technique that has been developed in the last 20 years and has shown to be an efficient tool for measuring the response function in a wide variety of materials [1]. The technique essentially consists of using visible or near infrared short-pulsed radiation to generate and coherently detect THz pulses [2-4]. The availability of ultra-short optical pulses allows for the emergence of nearly single-cycle THz radiation of (sub) picosecond duration [5-7]. The ability to set and control a delay between an optical excitation pulse and the subsequent THz probe pulse introduces the possibility of time-resolved measurements, with resolution determined by the duration of the THz pulse. Another appealing feature of this technique is the ability to directly measure the electric field of the THz radiation.
Thus, upon passage of the THz pulse through a sample, one can easily extract the complex response function of the material in a single measurement.
THz-TDS has been extensively used in the study of a wide-spectrum of materials: solid-state materials, polar and non-polar liquids, biological media, superconductors, nanomaterials, etc [8, 9]. Many interesting materials contain
12 important information lying in the THz frequency regime, such as vibrational modes in solid-state and organic materials, relaxation processes that occur on the picosecond time-scale, etc. In addition, the THz radiation is not ionizing, like x-rays for example, and is thus of great importance for biological imaging.
Significant efforts have been concentrated on continuously improving the characteristics of this technique by searching for novel THz generation and detection materials. The aim of this introduction Chapter is to give a brief description of the typical sources and detectors of THz radiation, the types of information that can be obtained by THz-TDS and the applications that the method has found. The emphasis would be on describing a ‘standard’ THz spectrometer which we use in the experiments presented further on in this thesis.
1.1 Generation and detection of THz radiation
The earliest work in the THz frequency range employed arc lamps as the source of radiation, while bolometers were used for the detection [10]. The limitations are apparent: the lamps are sources of continuous wave (cw), incoherent radiation of low brightness, while the detectors are only sensitive to the power. Improvement of technology has led to the development of new sources in the far-infrared (FIR): radiation can be obtained directly from fixed-frequency
FIR lasers or by frequency-mixing of two IR or near-IR lasers [11, 12]. This resulted in high-intensity radiation that was still cw in nature and of very limited
13 tunability. Recently developed systems such as synchrotrons and free electron lasers achieved intense, bright, tunable and pulsed (~3 to 10 ps) THz radiation, but such systems are very big and expensive [13]. This explains the advantages of THz-TDS based on ultra-short optical pulses: it is a table-top experiment, allowing for coherent detection of (sub) picosecond THz pulses, capable of time-resolved study in the FIR.
The first method for generation and detection of THz pulses is based on the use of photoconductive antennas [14-18]. Optically-generated carriers are accelerated in the presence of an applied bias voltage, resulting in emission of a burst of THz radiation. The commonly used semiconductors are low-temperature-grown GaAs and ion-implanted Si. The duration of the THz pulse is limited not only by the width of the excitation laser pulse, but by the response time of the material as well, since this is a resonant method. Thus THz pulses with duration of a few hundred femtoseconds were achieved. The ability to ‘capture’ this radiation with the help of a silicon lens and/or off-axis parabolic mirrors allowed for free space propagation of THz waves and standard spectroscopic measurements. The detection of THz radiation consists of a semiconductor (usually the same as the emitter) synchronously gated by an optical pulse, so that the current flow through the detector is proportional to the strength of the THz field present at that time. Adjustment of a variable delay line allows for a time-domain mapping of the THz electric field. A great deal of study
14 has been devoted to improve and optimize the performance of photoconductive antennas [19, 20].
When an amplified laser system is used a direct application of laser pulses onto a standard photoconductive antenna will result in damage due to the high intensity focused in a small spatial volume. One possible solution is to use large-aperture antennas where the separation between the electrodes is much larger, allowing non-focused radiation to be employed [21]. The problem of applying very strong bias voltage can be overcome and some of the highest THz pulse energies have been achieved using this method [22]. An internal electric field produced in the depletion layer of a semiconductor can also be used to generate THz radiation instead of the bias field. Application of a strong magnetic field in this geometry [23-26] has shown to result in a significant enhancement of the THz emission by increasing the radiation coupling efficiency of the photoinduced carriers in the depletion layer.
The most popular method for achieving THz radiation using an amplified laser system has been optical rectification, a second-order nonlinear process [27, 28].
A typical THz-TDS spectrometer, similar to the one we used in the experiments presented in this thesis, is shown in Fig. (1.1) [29, 30]. The system is based on a regeneratively amplified Ti: Sapphire laser, operating at 1 kHz repetition rate, producing 50 fs pulses at 800 nm, with ~ 0.8 mJ pulse energy.
15 Beamsplitter
Ti:Sapphire Amplifier, 1kHz Chopper Emitter 50 fs, 800mW
THz beam
Polarizer Sample
Probe delay Detector
Pump delay λ/4 waveplate Polarizing beamsplitter
Fig. 1.1 Standard THz-TDS spectrometer, based on an amplified laser system. THz radiation
is generated through optical rectification and detected through electrooptic sampling in ZnTe.
Generation of FIR radiation is obtained through optical rectification, a process described by the χ()2 nonlinear susceptibility [31]. It consists of mixing light at
frequencies ω1 and ω2 to produce radiation at the much lower frequency
ωω12− . This nonlinear interaction is achieved in a 1-mm-thick (110) ZnTe crystal. The availability of ultrashort optical pulses allows for the generation of broadband THz radiation, obtained by subtracting different frequencies from the spectrum of the fundamental [7]. The resultant pulse is centered at 1 THz and has similar bandwidth. In the time-domain this results in a nearly single-cycle pulse of duration ~1 ps. Optical rectification is a non-resonant method hence the
16 duration of the THz pulse is not determined by the response time of the material, but only by the bandwidth of the driving optical pulse.
THz radiation is detected through the process of free space electro-optic (EO) sampling [32] in a 2-mm-thick (110) ZnTe crystal. The method is based on the
Pockels effect in which an applied electric field induces birefringence in the EO crystal. For THz detection it is the presence of the transient THz electric field that acts as a bias field. Thus a co-propagating ultrashort probe pulse experiences a rotation of its polarization that is proportional to the magnitude of the THz field. By scanning the delay between the two pulses we can effectively map out the waveform of the THz radiation, measuring directly the electric field rather than the intensity. EO sampling is a non-resonant technique therefore the detection bandwidth is ultimately limited by the duration of the optical probe pulse.
The detection technique presented above relies on sampling of identical THz waveforms. It is possible to map out the form of the THz electric field in a single-shot measurement without relying on a delay line. One possible way is to introduce the THz and optical probe beams at an angle inside the EO crystal [33].
Thus different spatial parts of the optical pulse effectively sample the THz pulse at different times. Another possibility is to use a chirped optical pulse in combination with a monochromator and a CCD camera: different frequency components arrive at different times, sampling the whole THz pulse [34, 35].
The signal is collected by a pair of balanced photodetectors and the lock-in
17 technique allows us to perform measurements with signal-to-noise ratio of ~ 104.
A significant improvement is obtained by enclosing the space into which the THz beam propagates and purging the air, in order to avoid absorption and reemission of THz radiation from water vapor molecules.
Both optical rectification and EO sampling rely on efficient interaction of the generating and the radiated fields inside the nonlinear medium of finite thickness.
The conservation of momentum, referred to as phase matching, imposes constraints on the effectiveness of this coupling. Nahata et al. [29] have shown that for co-propagating waves the phase-matching condition requires the phase velocity of the THz wave to match the group velocity of the optical pulse.
Semiconductors such as GaP and GaAs, along with ZnTe, have been recognized as favorable nonlinear media. The phase-matching condition is critical for obtaining large bandwidth in the processes of generating and detecting THz radiation. Very short (10 – 20 fs) generating optical pulses combined with the use of ultra-thin (tens of microns) emitters and detectors have been used to produce extremely short THz pulses (bandwidth ~ 50 THz) [36].
The THz spectrometer presented above allows for the following measurements: placing a chopper on the probe arm and blocking the pump beam allows us to simply scan the THz waveform and obtain purely spectroscopic information. Moving the chopper to the pump arm and fixing the pump-probe delay time gives us the pump-induced change in the THz waveform, which can be
18 immediately related to the complex response of the investigated material at a fixed time after photoexcitaion. Finally, adjusting the delay between the THz pulse and the optical pulse used to measure it, at a position corresponding, for example, to the peak of the THz waveform, and scanning the pump-probe delay, yield time-resolved information with time resolution determined by the duration of the THz probe pulse, ~1 ps. Thus we see that this spectrometer allows us to perform a 2D scan of the investigated system, extracting spectroscopic information at variable delay time between the excitation and the probe pulses.
Lastly, we note that in all measurements we performed the optical pump beam was introduced in the setup through a small hole drilled in the parabolic mirror, ensuring it is collinear with the THz probe beam at the position of the sample.
This ensures the best temporal resolution since a non-collinear propagation introduces a time-delay between the excitation from different parts of the pump beam.
1.2 Applications of THz-TDS
THz time-domain spectroscopy has been used extensively to perform spectroscopic and time-resolved measurements in various material systems [9].
These include bulk semiconductors [37-41], liquids [42, 43], gases [44]. The dynamics of optically induced carriers was studied in various solids [39, 45-47], in low-dimensional structures, such as quantum wells (QWs) [48], and InAs/GaAs
19 quantum dots [49]. Nonequilibrium charge transport can be monitored by recording the emitted THz radiation of photoinduced carriers (THz emission spectroscopy). Using this technique, mobilities and charge transport properties were measured in various semiconductors [50, 51], dye solutions [52], QWs [53], high-temperature superconductors [54]. Transient conductivity in functionalized molecular crystals [55] was observed. Polarizability of isolated nanocrystals was measured using freely propagating THz pulses [56]. THz-TDS was used in studies of various superconductors [57, 58], as well as colossal magneto-resistance manganites [59]. Terahertz waves have been demonstrated in near-field imaging [60, 61].
1.3 Outline of the thesis
In the first part of this thesis we investigate some applications of the THz-TDS technique. In Chapter 2 we study the generation of THz radiation through a nonlinear process from highly-localized emitters, which is promising for possible near-field sub-wavelength imaging applications. Chapter 3 deals with the general problem of the finite size of the beams in a typical pump/probe experiment, an issue for THz-TDS due to the inability to tightly focus the THz radiation, a consequence of the long-wavelength nature of the radiation.
In the second part we investigate the response of optically-generated excitons in various systems of semiconductor quantum dots (QDs), to an externally applied
20 THz electric field, concentrating mainly on size-dependent scaling laws, the dependence on particular material parameters, and on many-body effects.
Chapter 4 is devoted to a summary of the electronic structure of CdSe and PbSe
QDs, two material systems that will be studied, with the help of various band models. In Chapter 5 we investigate the dependence of the polarizability of excitons in isolated QDs on their size and on the material. In Chapter 6 we use
THz-TDS to study the dominant effect of Auger recombination in CdSe QDs in the presence of multiple excitons and compare to existing studies based on complementary techniques, and in Chapter 7 we investigate the collective response of photogenerated carriers in QDs to external electric fields, accounting for the many-body carrier-carrier Coulomb interaction. Finally, Chapter 8 is devoted to a size-dependent study of the two-photon absorption coefficient in semiconductor QDs, using the optical pump/white light probe technique.
21 Chapter 2
Localized THz Generation via Optical Rectification in ZnTe
Terahertz time-domain spectroscopy (THz-TDS) based on mode-locked lasers has recently attracted significant attention because of its wide range of applications in spectroscopy and imaging [1, 62]. With this technique, broadband radiation up to a few THz can typically be achieved through the use of a photoconductor [1] or by means of the non-resonant nonlinear response of a suitable material [28]. The spatial resolution of conventional THz imaging, [60,
61, 63, 64] however, is rather poor (several hundred microns) since in the far field
it is limited by the THz wavelength λTHz . The concept of near-field microscopy
[65] has been applied to overcome the diffraction limit in THz-TDS: sub-wavelength apertures (either physical [66] or optically-induced [67]) have been refined to achieve a resolution of about 7 μm [66] and sharp tips [68-71] have been introduced to allow for sub-micron resolution [68]. In these near-field methods, although spatial resolution is ultimately limited by the size of the aperture/tip, the realization of the ideal spatial resolution is often hindered by the detection sensitivity due to the sharp dependence of the signal on the aperture/tip size. For instance, the radiation power throughput through a sub-wavelength aperture of size a scales as a6 [72].
22 An alternative approach is to use highly localized THz emitters or detectors based on nonlinear optics [65, 73]. By using a tightly focused optical beam to generate/detect THz radiation in a thin nonlinear material and placing the sample
of interest close to it, a spatial resolution of ~ λTHz / 300 (for a typical THz radiation at 1 THz), corresponding to the diffraction limit of the optical excitation,
~ λopt , should be achievable using a conventional THz setup. This approach greatly simplifies the fabrication of the near-field THz probes and can also be well adapted for parallel detection, as has been demonstrated for conventional
(far-field) imaging [64]. Xu and Zhang [73] first studied optical rectification in a
GaAs emitter with an excitation of a size comparable to or smaller than the center wavelength of THz radiation using a bolometer. They observed a a2 dependence of the total THz radiation power at small sizes under a fixed excitation power, where a is the size of the excitation. Since in most near-field techniques the radiation throughput often limits the achievable resolution, it is essential to obtain a deeper understanding of the dependence of the radiation on the size of the emitter.
In this Chapter we present a detailed study of the behavior of a highly localized
THz emitter based on optical rectification in ZnTe both experimentally and theoretically. Our result demonstrates that, for excitation sizes smaller than the
THz wavelength, the radiation throughput under a fixed excitation power from a thin, non-resonant, second-order nonlinear material is mainly independent of the
23 size of the excitation. This characteristic radiation throughput indicates that the aperture-less approach of near-field THz microscopy based on nonlinear optics can be advantageous when a spatial resolution of a micron or larger is required.
Based on a simple model and numerical simulation we show that this behavior of localized THz emitters is a combined effect of optical rectification and diffraction.
The previously observed a2 dependence [73] cannot be accounted for by diffraction and it is more likely due to higher-order nonlinear effects.
2.1 Difference-frequency generation
Difference-frequency generation (DFG) is an attractive tool for obtaining tunable radiation in the infrared part of the spectrum. From the standpoint of a typical nonlinear optics analysis DFG is very similar to the case of sum-frequency generation, with the notable difference that in the case of long-wavelength generation in the far-infrared, diffraction of the generated radiation must be taken into account. In the limiting case of near infrared generation, when the wavelength of the generated radiation is still much smaller than the transverse dimension of the beam, we can use the plane-wave approximation. For a fixed
excitation power Popt , since optical rectification is a second-order nonlinear
process, the nonlinear polarization is proportional to the optical intensity Iopt and
−2 scales with the lateral size of the excitation a as PNL∝∝IPa opt opt . For large
24 emitters ( a > λTHz ) we can readily use the familiar result from the plane-wave approximation:
2 222 − ETHz∝∝aP opt a (2.1.1)
The radiation power at all THz frequencies decreases as a−2 . In this case the size dependence of the emission is dominated by the nature of the optical rectification.
When the wavelength of the generated radiation approaches the far-infrared and becomes comparable to a , we have to take into account the diffraction effects. For a localized emitter of lateral size a and thickness l (20 μm used
here), a , l λTHz , the entire excitation volume adds constructively and we find the radiation power in the far field as
2 202 ETHz∝ alP NL ~a (2.1.2)
Here the a−2 dependence from the nonlinear interaction is canceled by the a2 dependence from diffraction. The radiation power is, thus, independent of the lateral size a of the localized source, in agreement with our experimental observation. The outcomes of the above simple model of optical rectification and diffraction are also in good agreement with the earlier results of Morris and
Shen [74] for difference-frequency generation.
We note that for an aperture-less localized THz emitter, studied both in this work and previously, [73] the result of diffraction from a sub-wavelength aperture of size a on a thin perfectly conducting screen, [72] for which the transmitted
25 electric field scales as a3 , is not applicable. Here diffraction contributes a factor of a2 in the radiation field in the far field. The previously observed a2 dependence of THz radiation on the excitation size, therefore, cannot be explained by diffraction.
2.2 Experimental setup
The details of the experimental setup for the THz-TDS have been described in Chapter 1 (see Ref. [28] as well). We used a Ti: Sapphire oscillator producing
100 fs pulses at 92 MHz instead of the amplified system, due to the extremely high intensities achieved when the excitation size is small, that can possibly damage the emitter. THz radiation was generated through optical rectification in a (110) ZnTe crystal of 20 μm thickness on a 0.5-mm-thick (100) ZnTe substrate.
A pump power up to 250 mW was used. The emission was collected by a pair of off-axis parabolic mirrors with a collection angle of 30° and was detected through electro-optic sampling in a 2-mm-thick ZnTe crystal. The excitation beam was
focused by an fl = 75 -mm lens normally on the emitter (a schematic drawing is shown in Fig. (2.2.1)). By translating the lens the size of the excitation a
(FWHM), was varied from ~10 to 350 μm. The size was first measured without the crystal in place using a knife edge, and then corrected numerically by taking the dielectric property of ZnTe into account. Note that for the excitation sizes of interest here, the diffraction of the optical beam within the emitter was negligible
26 and thus ignored. The curvature of the optical wave front was relatively small and its influence on the nonlinear interaction was insignificant. For each excitation size the electric-field waveform of the THz emission was recorded.
The spectral density ~ E( ω ) 2 was extracted through the Fourier transform of the time domain measurement.
Fig. 2.2.1 Sketch of the THz emitter employed in the measurement (not to scale): optical pump
beam (800 nm) is focused onto a 20 μm ZnTe emitter ( (110) orientation) on a 500 μm ZnTe
substrate ( (100) orientation). Translation of the lens allows variation of the effective size of
the emitter.
2.3 Results
Figure (2.3.1) illustrates the dependence of the spectral density of THz radiation on the size of the optical excitation a ; the excitation power was fixed.
Emission at three representative frequencies (0.9, 1.4 and 1.9 THz), normalized to their maxima independently, are depicted in the main panel.
27 1.4 0.9 THz 1.4 THz 1.2 1.9 THz
1.0
0.8 (a.u.) (a.u.) a (μm) 2 0.6 0 20 40 60 80 100 1.1 |Ε(ω)| 0.4 100% 1.0 50% 0.9 0.2 0.8
0.7 0.0 0.6
0 100 200 300 a (μm) 2 Fig. 2.3.1 Normalized spectral density E( ω ) versus the FWHM of the optical excitation a .
Three representative frequencies are shown in symbols. Solid line is a fit to Eq. (2.4.16).
Inset: size dependence of the total THz power (normalized) in the regime of two-photon
absorption using 100% (filled) and 50% (open) of the available excitation power.
Three distinct regimes can be identified: at very small excitation sizes (below ~ 30
μm), THz emission increases dramatically with increasing excitation size, an identical behavior for all THz frequencies. The emission then becomes almost independent of the excitation size (~ 30 - 150 μm) until it reaches a characteristic
value ac ; a decrease of the emission with increasing size follows afterwards.
The characteristic value of ac , on the order of a few hundred microns, shifts towards larger sizes for radiation of longer wavelengths. To be more qualitative
28 we define ac to be the size of the emitter for which the emission decreases by
10% from its maximum value. The dependence of ac on the emission
wavelength λTHz is shown in Fig. (2.3.2) as symbols. It is seen to increase monotonically with the THz wavelength.
350
300
250 m)
μ 200 ( c a
150
100
50
50 100 150 200 250 300 350 λ (μm) THz
Fig. 2.3.2 Characteristic size of the emitter ac as defined in the text for the transition of
0 −2 spectral density behavior (from a to a dependence) versus the THz wavelength λTHz .
Symbols: experiment; line: simulation.
We compare the size dependence of the THz emission generated in ZnTe with
100% and 50% of the pump power. As shown in the inset of Fig. (2.3.1), the size dependence with 50% pump power is clearly less steep than the 100% case
29 although the diffraction effect should be similar in both cases. The result indicates that other physical mechanisms such as higher-order nonlinear processes induced by high excitation intensities achieved at very small sizes in the emitter are responsible for the observed trend in the THz emission.
2.4 Numerical simulation and discussion
For a more quantitative comparison we perform a numerical study of the THz generation via optical rectification from a thin, non-resonant second-order nonlinear medium and THz propagation in free space. We have adopted a model developed by Côté et al. [75]. In the model, the THz radiation is represented by a superposition of plane waves of varying angular and lateral spatial frequencies with appropriate boundary conditions for each component.
We model the radiation source as an instantaneous second-order nonlinear polarization, induced by a non-depleted optical pump beam with a Gaussian temporal and spatial profile. The plane-wave approximation is used for the optical pump beam, which is valid for sufficiently thin crystals, in which the diffraction of the optical beam is negligible. The finite collection angle of the optics is accounted for by integrating the calculated radiation density over an area at a distance from the emitter corresponding to the experimental geometry. In this Section we will present a brief summary of this theoretical model, illustrating the main points of interest.
30 The problem can essentially be separated into two parts, the first one treating the problem of finding the electromagnetic radiation resulting from a given source in free space, while the second part deals with the light propagation through a given layered medium. The importance of handling the generation and propagation of THz radiation in such detail is due mainly to the long wavelength of the emitted radiation. From one point of view it becomes comparable to the dimensions of the generating medium with the presence of interfaces having increased importance, and from another the strong diffraction of the THz beam following a short propagation distance cannot be treated analytically with the help of a governing equation, similar to the parabolic equation for optical waves.
In free space, the set of Maxwell equations leads to the wave equation for the electric field, which for monochromatic light has the form
(∇222+=Ω n ) Er() 0 (2.4.1) and a similar equation for the magnetic field, where ∇2 is the Laplacian operator,
ΩΩ = /c, where Ω is the angular frequency, c is the speed of light in vacuum, and n is the index of refraction, which is a complex function of frequency.
Since any real electromagnetic field can be represented as a sum of plane monochromatic waves we shall concentrate on treating such waves, bearing in mind that an inverse Fourier transformation can always reconstruct the radiation in the space-time domain. Assuming plane wave solutions
31 Er( ) = Eexp( iν ⋅r ) (2.4.2) leads to the following condition for the wave vector
ν 222= Ω n. (2.4.3)
Since one of Maxwell’s equations stipulates that in this case the electric field and the wave vector are perpendicular to each other, we can decompose the wave vector as
ν = Kz± w, ± (2.4.3) w =−Ω 22nK 2 where K is the component in the (x,y ) plane, and the ± sign accounts for the possibility of forward and backward (with respect to the z direction) propagating waves. Here, we can introduce a new set of unit vectors that can be used to decompose the wave vector:
sKz= × , (2.4.4) pzK± = (K∓ w )/ν
These are the s − and p − polarized components of the electric field, that are especially advantageous in treating light propagation through different media, because of their independence from each other, manifested through the Fresnel coefficients that incorporate the boundary conditions. Then, the form of a, for example, forward propagating wave can be written as
Er+++++()= (Esp s+⋅ E p) exp( iν r ) (2.4.5) where the electric field components are to be determined from a specified source of radiation and a given geometry. So far we have treated the case of a
32 homogeneous wave equation, describing the propagation of radiation in free space. In the presence of a source of radiation the electric field can be found with the help of the Green’s function formalism. Given a Fourier decomposition of a given polarization source
dΩ dK Pr(,t)= P (Ω, K ,z )exp( i K⋅− R iΩt ) + c.c., (2.4.6) ∫∫2()ππ2 2 the radiated electric field is obtained from the following Green function integral EK()Ω,,z=−⋅∫ G(Ω,;zz' K) PK(Ω,,z'dz') (2.4.7) where for the case of an infinite medium with dielectric constant ε the Green’s function has the form 21− G ()Ω,;zK =+2π iΩ w(ss p++ p )θ ( z) exp( iwz) + 21− 2π iΩ w()()()ss+ p−− p θ −−− z exp iwz (2.4.8) 4πε−1zzδ() z where θ ()z is the Heaviside function, δ( z) is the Dirac function, and the polarization vectors are written in dyadic form. This completes the first part of our task.
The problem of light propagation through layered structures with differing dispersion properties is discussed in numerous texts [76], therefore we shall present only the most common results, applicable to our case. Since in general in a given layered medium radiation will be propagating both in forward and backward directions, it is useful to represent the electric field in the form
33 ⎡ Eii+ exp( iw z ) ⎤ ⎢ ⎥ (2.4.9) ⎣Eii− exp(− iw z )⎦ where the index i labels the medium. Then in the presence of a single interface the electric field on both sides of the boundary can be found with the help of the transfer matrix
1 ⎡ 1 rij ⎤ M = (2.4.10) ij ⎢r 1 ⎥ tij ⎣ ij ⎦ where rij and tij are the Fresnel coefficients, differing for the particular light polarization. Propagation through distance z can be expressed as
⎡exp( iwi z ) 0 ⎤ M(z)i = ⎢ ⎥ (2.4.11) ⎣ 0 exp(− iwi z )⎦
D
n n n1 2 3
Fig. 2.4.1 Typical layered geometry utilized in our experiment
The geometry of our experiment, Fig. (2.4.1), is characterized by the following transmission and reflection coefficients:
t13 t 32 exp( iw 3 D) t 13 r 32 t 31 exp(2 iw 3 D) T,12==+ Rr 12 13 (2.4.12) 12−−r31 r 32 exp() iw 3 D 12 r 31 r 32 exp() iw 3 D
34 where the Fresnel coefficients can be written for the s − and p − polarized light.
Having established the general outlines for the treatment of generation and propagation of electromagnetic radiation in a layered structure, we can consider
Eq. (2.4.7) and (2.4.12) to obtain the following expression for the electric field propagating in the z direction:
⎡ D EKΩ,;z=⋅−+2π iΩ 21w− Cq exp iw zqq dz' exp iw z' PΩ,;z' K +++()31133∑ ()⎢ ∫ ()() q ⎣ 0
D ⎤ rq exp2 iw Dqq⋅ dz' exp iw z' PΩ,;z' K 32() 3 1+− 3∫ ()() 3 ⎥ 0 ⎦
(2.4.13)
q q t31 C = qq (2.4.14) 12− r31 r 32 exp() iw 3 D where q stands for the s − and p − polarizations.
Thus representing the polarization as a second-order nonlinear process, caused by a non-depleted optical pump beam with Gaussian temporal and spatial shape and plane wave front, we can obtain solutions for the THz electric field of a particular angular Ω and lateral K spatial frequencies at a given distance z away from the emitter.
Results of the simulation are shown in Fig. (2.4.2). In part (a) we illustrate the size-dependence of the spectral density for several representative THz frequencies collected within 30° in the forward direction.
35 3
2 (a)
1 -2 7 ∼a 6 5 4
3
2
0.9 THz 0.1
7 6 1.4 5 4 3 1.9 2 (a.u.) (a.u.)
2 3
2 (b) |Ε(ω)| 1.4 THz
1 -2 ∼a 7 6 o 5 30 4
3
2 o 15 0.1
7 6 5 o 4 7.5 3
2
4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 3 100 1000 a (μm)
2 Fig. 2.4.2 (a): Simulated spectral density E( ω ) at 0.9, 1.4, and 1.9 THz (solid lines) versus the FWHM of the optical excitation a for a 30o collection angle. (b): simulated spectral
2 density E( ω ) at 1.4 THz for collection angles of 30°, 15°, and 7.5°. Dashed lines illustrate
a−2 dependence.
36 It reproduces the trend observed experimentally: for small sizes of the emitter the radiation throughput is largely independent of the size a , but decreases as a−2
(dashed lines) for larger emitters. The transition size ac , extracted using the same criterion as for the experimental data, is shown as a solid line in Fig. (2.3.2).
The agreement between the results of the experiment and simulation is very good.
To explore the effect of the finite collection angle we also study the dependence of the radiation throughput on the size of the collection angle. The excitation size dependence of the THz emission at 1.4 THz for three representative collection angles (30°, 15°, and 7.5°) is illustrated in Fig. (2.4.2 (b)).
All cases show an identical trend for the size dependence as described above.
The detection efficiency, however, decreases rapidly and the transition of spectral density behavior (from a0 to a−2 dependence) shifts to larger sizes with decreasing collection angle. This can be understood as a consequence of diffraction: radiation from very small emitters diffracts significantly and exceeds the range of the collection optics. The maximum detectable radiation is thus, in part, limited by the collection optics for small emitters.
As we have discussed so far, the reduction of the radiation throughput at very small excitation sizes cannot be explained purely by diffraction (see Fig. (2.3.1)).
It, however, is likely to arise from higher-order nonlinear processes such as two-photon absorption (TPA) of the optical excitation and enhanced
37 photo-induced carrier screening of the generated THz emission in the emitter. In the current measurements of THz emission from a thin ZnTe emitter on a substrate the optical radiation encounters the ZnTe substrate of thickness
L = 500 μm prior to the l = 20μm ZnTe emitter. TPA in the substrate, therefore, can be significant to cause an appreciable loss of excitation energy, being a dominant mechanism for a reduction in the THz emission. Variation of the excitation intensity in a crystal of negligible linear absorption can be described as
Iopt ()0 I(z)opt = (2.4.15) 10+ βIopt ()z where z is the distance of propagation in the crystal and β is the TPA coefficient. The THz emission from a sub-wavelength source of Eq. (2.1.2) including TPA now reads as
2 2 2 Popt E~alTHz 2 (2.4.16) a + βPLopt
A fit of the experimental data to Eq. (2.4.16) yields a TPA coefficient of β ∼ 2 cm/GW for ZnTe, which is of the same order of magnitude as given in the literature [77].
In the earlier study (Ref. [73]) the reduction in the THz emission is likely due to the enhanced screening of THz radiation by carriers injected in GaAs by the above-band-gap optical pump radiation (820 nm). From the intensity dependence of THz emission from GaAs through optical rectification, measured with a similar optical pump, [78] we estimate the saturation pump intensity to be
38 on the order of 100 MW/cm2. This saturation intensity corresponds to an excitation size of ~ 100 μm in the measurement in Ref. [73], below which a significant reduction in the THz emission was observed.
2.5 Conclusions
We have examined the radiation throughput of a localized THz emitter based on optical rectification in a thin, non-resonant second-order nonlinear medium both experimentally and numerically. For a fixed excitation power the nature of the second-order nonlinear interaction gives rise to a a−2 dependence on the excitation size of the THz emission from a large emitter. The radiation throughput remains constant when the excitation size becomes smaller than a
characteristic size ac , being on the order of the THz wavelength. This is a combined effect of the second-order nonlinear interaction (radiated field ∼ a−2 ) and diffraction ( ∼ a2 ). This favorable throughput characteristic of the sub-wavelength THz emitter is promising for use in near-field THz microscopy,
which can potentially allow for a spatial resolution of λTHz / 300 (for radiation at 1
THz) with a conventional THz setup, limited by diffraction of the optical excitation.
Another important consideration for aperture-less near-field THz microscopy based on nonlinear optics is the choice of the nonlinear medium. A strong reduction of the size of the optical excitation limits the amount of the usable excitation power since higher-order nonlinearities (such as TPA, photo-induced
39 carrier screening, and photo-damage) become increasingly relevant. For THz generation via bulk nonlinearities such as employed in this study the medium has to be much thinner than the THz wavelength and the Rayleigh length of the optical excitation beam. The use of a surface nonlinearity in a reflective geometry may simplify constraints on the emitter.
40 Chapter 3
Finite Pump Beam Size Effects in Optical Pump/THz Probe
Spectroscopy
Optical pump/terahertz (THz) probe spectroscopy has emerged as an effective technique for the study of charge transport in condensed matter systems
[9]. In this method the complex conductivity, induced by optical excitation, is probed over a broad spectral range by a THz pulse at a controlled delay. The spectroscopic information is obtained from the spectrum of the complex conductivity and the dynamics is inferred from the dependence of the conductivity on the pump-probe delay time. An important experimental issue in pump/probe measurements, particularly optical pump/THz probe, concerns maintaining a homogeneous photoexcitation over the area of the probe [8, 30]. In an ideal measurement the pump beam size would far exceed that of the probe beam and thus easily ensure homogeneity of excitation of the sample. In the case of optical pump/THz probe, however, it is rather difficult with a limited pump power to generate homogeneous excitation over the THz probe area due to the large spot size at its focus, a reflection of the long wavelength nature of the THz radiation.
The consequences of a varying excitation density over the THz probe volume, as we demonstrate below, are not restricted to the obvious averaging of the
41 measured material response associated with the differing local excitation densities. Rather, one must consider spectral distortions that might be introduced by diffraction of the THz beam in propagating through the spatially inhomogeneous sample. Such possible waveform distortions may be further enhanced due to the variation of the spectral content of the broadband THz probe across the transverse profile of the beam [66, 79, 80]. In this Chapter, we model the optical pump/THz probe signal as radiation from a pump-induced current driven by the THz probe field and we derive, for the first time, a requirement on the pump beam size for a readily interpretable measurement.
3.1 Theoretical model
The pump-induced change in the THz probe electric field can be treated as radiation from the photo-injected carriers driven by the THz electric field. The size of the radiation source is thus determined by the overlap of the two beams.
For simplicity we assume a Gaussian profile with a FWHM of apump for the pump
intensity and aprobe for the probe electric field. The size of the source a can be
22 2 expressed as 11/a=+ /apump 1 /a probe for a one-photon pump-induced process.
In the previous Chapter we investigated the dependence of the THz emission on the lateral size of the source. When the source polarization is kept constant, the radiation power scales with the size of the emitter a as a4 for small emitters and as a2 for large emitters. The transition from an a4 to an a2 dependence
42 is a consequence of the superposition of radiation from subparts of the source separated by a wavelength or more, that are no longer perfectly in phase. The
characteristic dimension ac for the transition is on the order of the wavelength and dependent on the details of the experiment.
Similarly, in optical pump/THz probe measurements, when the pump fluence is fixed, the pump-induced change in the THz probe electric field also consists of two major regimes, as shown in the simulation of Fig. (3.1.1). Using the above scaling law, we can derive the following dependence of the induced THz signal,
normalized to that of an ideal measurement ( apump →∞), on the ratio of the two
beam sizes α = apump /a probe :
2 2 2 ⎛⎞α ΔE(normω )= ⎜⎟2 , a probe< a c (3.1.1) ⎝⎠1+ α
α2 = , a> a (3.1.2) 1+ α2 probe c
In both regimes the pump-induced signal increases nonlinearly with the size of the pump and saturates at the value of 1 by definition for large α' s, corresponding to the ideal measurement.
43 1 10 2 ∼a ac 0 1.15 THz 10 0.45 THz
-1 10
-2 (a.u)
10 2
-3 10 |ΔΕ(ω)| 4 ac -4 ∼a 10
-5 10
-6 10 7 8 9 2 3 4 5 6 7 8 9 2 3 4 5 6 0.1 1 Radiation source size (mm) Fig. 3.1.1 Simulation of the radiation power at 0.45 (dashed line) and 1.15 THz (solid line)
emitted from a source of a fixed peak polarization but varying sizes. The radiation is collected
by off-axis parabolic mirrors of 64 mm in diameter and 120 mm in focal length as used in the
4 2 experiment. Critical sizes ac for the transition from the a to the a dependence are indicated
for both frequencies.
A criterion for the required pump beam size in an ideal measurement, therefore, can be readily obtained from Eq. (3.1.1) and (3.1.2). For instance, to ensure 10% or less departure from the idea measurement the ratio of the pump
and probe beam sizes has to be α ≥ 43. (for aaprobe< c ) or α ≥ 3 (for aaprobe> c ).
In the region of aaprobe< c the signal approaches the ideal measurement slowly due to the strong diffraction effects. The spectral content of the THz pulse at its focus often varies across the transverse profile: lower frequencies usually have
44 larger foci due to diffraction. To take the size variation into account we would use the size of the lowest THz frequency that would be used to interpret the
spectrum as aprobe and determine the required size of the pump accordingly.
Because of the saturation behavior, the spectral shape distortions are expected to be insignificant.
3.2 Experimental setup
We employed a conventional collinear optical pump/THz probe setup, similar to the one described in Chapter 1. Thin samples of single crystal ZnTe or CdTe
(0.5-1 mm) were placed at the focus of the THz probe beam which was 1.3 mm in
FWHM of its peak electric field in the time domain. An optical pump at 800 nm was used to inject carriers through two-photon absorption in ZnTe and one-photon absorption in CdTe. The size of the pump beam was varied from 0.5 to 3.6 mm (FWHM) and the peak fluence was kept at a constant value. The pump-induced change ΔE(t) in the transmitted THz electric-field waveform
E(t) was measured at a sufficiently long delay (200 ps) after the photoexcitation to exclude the hot carrier effects, but to remain in a regime where carrier recombination is insignificant. The spectral dependence of the pump-induced conductivity σ( ω ) =+σ′′′( ω )iσ ( ω ) was then extracted through Fourier transformation of the electric-field waveforms [30]. We have chosen the pump fluence such that the maximum modulation in the THz field for all measurements
45 did not exceed 15% and could be treated as a small perturbation.
3.3 Results and Discussion
The complex conductivity over a spectral range of 0.3 – 1.6 THz induced by a pump beam of 0.7 and 3.6 mm in CdTe at room temperature is depicted in Fig.
(3.3.1 (a)). The result obtained with the larger pump beam (3.6 mm, solid lines) is seen to be described well by the simple Drude model (grey lines),
−1 σ( ω )ne=+2τ ()mi* ()1 ωτ , where n , τ and m* are the carrier density, scattering time and effective mass, respectively.
40 σ' (a) (b) 4 30 3.6 mm 10 0.7 mm |σ(ω)| 3 20 10 m) 2 Ω
2 (1/Ω 10 10 '' (1/ m) σ 0 1 '+i 8 10 2 σ 6 σ'' 1.15 THz 0 4 0.45 THz 10 2 -1 0 10 12 0.4 0.8 1.2 1.6x10 0 1 2 3 4 Frequency (Hz) Pump size (mm)
Fig. 3.3.1 (a) Complex conductivity of photo-induced carriers in CdTe at room temperature as a
function of frequency for pump sizes of 3.6mm FWHM (solid lines) and 0.7mm (dashed lines).
Fit to the Drude model is shown for 3.6mm pump (grey lines). (b) Pump size dependence of
2 σ( ω ) at 0.45 (empty circles) and 1.15 THz (filled circle). Lines: fits to Eq. (3.1.1).
46 Using the electron effective mass m.m* = 013 [81] we extract the electron mobility μ = eτ /m* to be ~ 520 cm2/Vs in CdTe, which is in a good agreement with literature [82]. Note that the hole contribution to the conductivity is neglected in the analysis since the hole mobility is an order of magnitude lower than the electron mobility [82].
The conductivity measured with a smaller pump beam (0.7 mm, dashed lines), however, is significantly different: overall it is about 2 times smaller in amplitude and can no longer be described by the Drude model. A detailed trend of the pump size dependence of σ( ω ) 2 is illustrated in Fig. (3.3.1 (b)) for two representative frequencies 0.45 and 1.15 THz. The induced conductivity increases nonlinearly with increasing the pump size, but at differing rates for different THz frequencies, which leads to spectral shape distortions.
It is evident from this example that information from measurements made under conditions far from ideal cannot be correctly extracted without a priori knowledge of the optical setup. The dependence of Eq. (3.1.1) is shown to reproduce the measurement of Fig. (3.3.1 (b)) very well. The fits (solid lines) correspond to a THz size of 1.4 and 0.9 mm for 0.45 and 1.15 THz, respectively, which agrees well with that of a direct measurement.
In practice, however, it is often challenging to fulfill the requirements for an ideal optical pump/THz probe measurement. Nevertheless, certain reliable information can still be extracted. As an example, we analyze the conductivity of
47 Fig. (3.3.1) over the range of 0.3 -1.6 THz measured with a pump beam of varying sizes using the Drude model. The extracted parameters including the mobility and the carrier concentration are illustrated in Fig. (3.3.2). The quality of the
Drude fit decreases with decreasing the pump size and the results obtained with
pumps of a.pump ≤17 mm can no longer be described by the Drude model and the inferred parameters (marked with ×) are not reliable.
15 5x10 (a) (b) 1000 4 800 μ ) 3 (cm -3 600 2 2 /Vs)
n (cm 400
1 200
0 0
0 1 2 3 4 0 1 2 3 4 Pump size (mm) Pump size (mm)
Fig. 3.3.2 Concentration (empty circles) (a) and mobility (empty squares) (b) of photo-induced
carriers in CdTe inferred from the fit of the measurement to the Drude model over a range of
pump beam sizes. Data that cannot be described by the Drude model are indicated by a ‘x’;
line (a): fit of the extracted density to the model as described in the text; (filled circles) (a): peak
carrier density inferred from our model.
48 For sufficiently large pump beams ( a.pump >17 mm) the mobility (squares, Fig.
(3.3.2 (b))), determined by the overall shape of the spectrum, has a relatively weak dependence on the pump size. The carrier density (empty circles, Fig.
(3.3.2 (a))), inferred from the magnitude of the conductivity, on the other hand, varies significantly. Its dependence on the pump size, however, is well described by Eq. (3.1.1) ( n~ ΔE( ω ) ) with an effective THz size of 1.7 mm (solid line, Fig. (3.3.2 (a))). Using the fit we can then infer the values for the peak carrier density (filled circle), which deviate no more than 10% with the pump beam size. This example demonstrates that from a measurement with a pump beam larger than a typical THz probe beam one can still infer the correct order of magnitude for the mobility and the peak carrier density.
3.4 Conclusions
In summary, we have demonstrated spectral distortions both in the amplitude and shape of the optical pump-induced conductivity probed by a THz pulse due to the inhomogeneous photoexcitation, a consequence of the finite size of the pump beam. A simple model that treats the pump-induced THz signal as radiation emitted by the photo-injected carriers driven by the THz probe field has allowed us to identify regimes of the pump size for which these distortions are either insignificant or correctable. For an ideal measurement (< 10% distortion in the spectrum) the pump beam has to be 3-4 times larger than the THz probe beam for
49 a one-photon excitation process. We have also been able to relax the requirement on the size of the pump for a readily interpretable measurement by introducing a metallic aperture at the focus of the THz beam to confine the probe area. The validity of the use of sub-wavelength apertures in THz measurements of samples of finite thickness, however, is unclear and requires further investigations. Diffraction and backscattering may be significant and make the interpretation of the measurements complex. The aperture also reduces the THz transmission, and can potentially introduce waveguide modes and cause pulse reshaping which usually leads to a degradation of the temporal resolution in the pump/probe measurements [66, 79, 80].
50 Chapter 4
Introduction to Semiconductor Quantum Dots
Quantum dots (QDs) are structures that confine the charge carriers in all three dimensions. This confinement is achieved by the presence of a boundary surface. In what follows we will be concentrating on semiconductor QDs, also called nanocrystals that are usually fabricated in the range of 1 – 10 nm in radius, thus containing between 102 and 105 atoms [83, 84]. The existence of a boundary leads to the appearance of discrete energy levels and the position of these levels is determined by the radius of the QD (Fig. (4.1)) [85]. The strong dependence of virtually all electrical and optical QD properties on its size opens the door for many investigations and possible applications [86]. For example, the band gap can be tuned by more than 1 eV by simply varying the size of the nanocrystal, thus an electron-hole recombination in QDs of the same material but with different radii leads to the emission of light of different color. In addition, due to the strong spatial confinement various many-body processes are significantly enhanced, as compared to bulk materials [87]. Quantum dots can be contrasted to other nanostructures such as quantum wires and quantum wells, which confine the charge carriers in one- and two dimensions, respectively.
51 6
5
4
3 Absorbance, a.u.
2 R = 3.33 nm 2.72 1 2.31 1.8 1.27 1.17 0 2.0 2.5 3.0 Energy, eV
Fig. 4.1 Linear absorption of colloidal CdSe/ZnS nanoparticles of varying sizes dissolved in
hexane.
The appearance of discrete energy levels and the relatively small number of electrons in QDs makes them similar to atoms. An important difference stems from the fact that the confined electrons do not move in free space, as in atoms, but in the semiconductor host crystal. This resembles the behavior of charge carriers in bulk materials; therefore the band structure of the host material plays an important role for the properties of the QDs. This simplified description of the nature of the quantum dots allows us to identify them as objects possessing both atomic and bulk material characteristics.
The progress in our knowledge of semiconductor nanocrystals is intricately linked to the ability to produce large quantities of stable QDs of controllable size
[88, 89]. In a typical experiment a size-deviation manifests itself as an
52 inhomogeneous broadening of the otherwise sharp energy peaks , which is certainly an effect that would much rather be avoided, since successful mapping of the size-dependent properties requires nanocrystal samples that are uniform in size, shape, structure and surface treatment. In the next Section we will briefly describe some of the more popular methods for fabrication and characterization of semiconductor nanocrystals that by now have become standard methods for obtaining large numbers of QDs.
4.1 Fabrication and Characterization
Among the bench-top methods the colloidal synthesis so far has produced nanocrystals of the best size uniformity [88, 90, 91]. This method involves mixing reagents in a vessel containing a hot, coordinating solution. A rapid increase of the temperature leads to super saturation of the species, which can be relieved by the process of nucleation. Upon the onset of nucleation the concentration of species decreases so that supersaturation cannot occur again and no nuclei can form. As a result of this the size of the samples only grows in time as more and more material adds to the existing nuclei. In general the size of the resultant nanocrystals can be controlled by the reaction time, the temperature and the chemistry of the reagents, while the size-distribution is mainly determined by the short period of time when nucleation occurs.
Stabilization of the newly formed QDs is achieved by adding suitable surfactants,
53 that effectively create an organic shell, also called a capping layer that binds strongly to the QD and prevents a further growth of the sample. In conjunction with size-selective precipitation this synthetic approach can produce samples with less than 5% standard deviation in size. Furthermore, selecting solvents with a particular polarity, such that upon evaporation of the solvent a weak van der
Waals attractive force appears and allows for the assembly of nanocrystals in the form of a superlattice, also known as a colloidal crystal [85, 92].
Self-assembled QDs nucleate spontaneously during molecular beam epitaxy, when a certain material is grown on a substrate which is not lattice-matched [93].
The resultant strain produces islands on top of a wetting layer, which can be subsequently buried to form nanocrystals. This method for growing QDs is known as the Stranski-Krastanov method and have been used to grow QDs of
InAs, InGaAs, InP, InSb, GaSb, etc QDs, usually on substrates of GaAs [94-98].
In addition, vertical stacking of QD layers is easily achieved, which effectively increases the density of nanocrystals and this has been used in the manufacturing of laser diodes, utilizing the QD layer as an active medium [99-101]. The drawbacks are the cost of fabrication and the lack of control of positioning individual QDs, as well as the relatively wide size distribution (around 15%). In addition, these epitaxially grown QDs have relatively large sizes, >10 nm, and are not appropriate for the investigation of size-dependent properties in the strong confinement regime.
54 Historically the first evidence of size-dependent properties of QDs was obtained by investigating semiconductor-doped glasses [102-106]. The method developed for obtaining nanocrystals involves adding a few weight percent of a semiconductor, usually Cd, Se, and S, into the batch materials of an otherwise transparent silicate glass [107]. Stress-free optical quality glass is produced by annealing, and subsequent heat treatment for a long duration (a few hours) is responsible for the formation of QDs. The duration of the heating is used to control the size and density of nanoparticles. This technique is relatively simple, and the glass samples exhibit rigidity and stability, but the nanocrystals have a large amount of surface defects, and relatively large, ~ 15%, size-distribution.
The uniformity of size, shape and internal structure of nanocrystals can be characterized with the help of various experimental techniques. High-resolution
Transmission Electron Microscopy (TEM) yields images revealing the internal crystal lattice, as well as the shape, of individual QDs (Fig. (4.1.1 a))). Lower magnification TEM images serve as a useful tool to quantify the size and shape dispersion of QDs within a given sample [91, 108] (Fig. (4.1.1 b))).
55
Fig. 4.1.1 a) High resolution TEM image of PbSe QDs, revealing the lattice structure; b) Low
resolution image, showing an ensemble of QDs [88].
Further information about the properties of the crystal lattice of the QDs can be obtained form Wide-Angle X-ray Scattering (WAXS). Measurements performed on various samples, such as Pb (Se, Te, S), Cd (Se, Te, S) QDs, etc., prepared by the method of colloidal synthesis, indicate that the diffraction patterns match those of the corresponding bulk materials. This is extremely important information for construction of theoretical models for the electronic structure of particular QDs.
Finally, self-assembled QDs in the form of a superlattice can be imaged with the help of TEM, showing the formation of ordered domains, while Small-Angle X-ray
Scattering (SAXS) measurements provide information for the spatial orientation of the nanocrystal superlattice [85, 89].
Above we have described only a few most common methods for fabricating and characterizing semiconductor QDs. Nanofabrication is a very active
56 inter-disciplinary area and many new techniques are being developed for high-quality samples of variety of materials. In this thesis we will be focusing on colloidal QD samples. As previously mentioned the state-of-the-art technique of colloidal synthesis guarantees the production of samples that differ by no more than one atomic layer from each other. The use of such excellent QD assemblies has helped to elucidate many uncertainties and ambiguities that have arisen from the natural sample size-distribution. We should also point out that in recent years it has become possible to observe and track single quantum dots for an extended period of time, thus observing single-particle properties that are otherwise buried in the ensemble measurements. These technological advances have proven to be essential for many biological applications. In the next Section we briefly summarize the applications that nanocrystals have already found.
4.2 Applications
Numerous applications of quantum dots arise from the following characteristics: exceptionally high quantum yield, approaching 90% in some materials, excellent stability over long periods of time, bright fluorescence, discreteness of the energy levels, flexibility achieved by precise control of the nanocrystals size, etc. Probably the most widely-used application which QDs have found is in biology and medicine, where the superb quality of the
57 fluorescence has been used to replace organic dyes as fluorophores [109-113].
For example, some of the earliest uses of QDs have been in the area of labeling fixed cells, tissues, etc. [114, 115]. Live-cell imaging of microinjected nanocrystals has been conducted in human cancer cells [116]. Quantum dots have been used in animal biology, obtaining images of targeted QDs in live mice, etc [117]. A great deal of effort in this field is devoted to reducing the toxicity that
QDs might introduce when injected in live organisms [118]. Improved techniques to create an effective capping layer that would preserve the quality of the QDs as well as reduce the availability of trap states, has led to a decrease of the role of blinking in single-particle experiments, and to the emission of sharp,
Gaussian-like fluorescence peaks that preserve their quality for extended periods of time. This phenomenon has led to the development of QD-LED, the use of nanocrystals as light-emitting diodes to make QD displays [119]. Theoretically the fluorescent characteristics of the QDs compared to the conventional LEDs indicate better image quality and more efficient energy conversion. Furthermore, the sharply defined energy states are promising candidates for use in quantum computation, where several entangled QDs, or qubits, can be used to perform quantum calculations [120]. In addition, quantum dots have shown much promise as a building block for a photovoltaic cell [121]. In materials such as
PbSe QDs, for example, it has been demonstrated that absorption of a high-energy photon can lead to the creation of multiple excitons via the process of
58 impact ionization [122, 123]. Various ways are being attempted to extract the electrons and holes before they recombine, a task that has been quite difficult due to the extremely fast Auger process that effectively reduces the number of electron-hole pairs.
The experiments described in this thesis aim at studying basic phenomena that occur in different systems of isolated quantum dots under different conditions.
The common factor in these experiments is the fact that the behavior of the investigated system is governed by the specific electronic structure of individual nanocrystals. The remainder of this Chapter is devoted to a detailed description of a few theoretical models that attempt to present a correct picture of the band structure in quantum dots made from CdSe and PbSe, the two systems that we concentrate on. In the last Section we discuss some of the limitations of the theoretical models we present.
4.3 Electronic structure of semiconductor quantum dots
A correct determination of the electronic structure of nanocrystals is of primary importance for gaining a deeper understanding of the various processes that occur in these systems. Most semiconductor QDs are synthesized in the range of 1 – 10 nm radius, meaning that on average there are between 102 and 105 atoms comprising every nanoparticle. Experimental measurements confirm that the atoms in nanocrystals are arranged in the form of the crystal lattice
59 corresponding to the bulk material. At the same time the percentage of surface atoms, especially for smaller QDs, represents a significant fraction of the total number of atoms, resulting in a strong dependence of many QD properties on the surface properties of the nanocrystal. Thus the determination of the electronic structure of QDs has to take into account both the underlying band structure specified by the particular crystal lattice (information that can be borrowed from bulk calculations) and the presence of a surface, which manifests itself as a potential barrier.
Historically the first attempt to model the quantization of electron and hole states was proposed by Efros et. al [124], Brus [125, 126], Rosetti et al. [102, 104].
This model ignores the crystal lattice structure of the QD and treats the system essentially as a ‘particle-in-a-box’ problem. The resultant energy levels are discrete and ‘memory’ about the crystal structure ‘survives’ in the assumption that the masses of the carriers are the corresponding effective masses in the bulk material. Despite its simplicity the model provides a meaningful tool for at least qualitative estimations, and allows for the important determination of different confinement regimes, based on the relative strength between the exciton’s kinetic and Coulomb energies. This model is described in detail in Section 4.3.1.
In reality, however, most semiconductors do not exhibit a simple parabolic dispersion relation due to the interaction of various bands. Probably the most widely used and successful model for calculating the band structure in the
60 materials of interest here, CdSe and PbSe, is the kp⋅ method. In Sections
4.3.2 and 4.3.3 we will present this model, which, in combination with appropriate boundary conditions, provides a very good description of the energy and size dependence of the quantized electron and hole levels.
4.3.1 Parabolic band model
In this simple model we will consider the quantum dot as a three-dimensional potential box in which electrons and holes are ‘created’ upon some excitation process, for example optical excitation. To obtain the possible states that these particles can occupy we will start with a review of some basic textbook results.
The Hamiltonian for a single particle in an infinitely deep spherical potential well U(r ) (U(r )= 0 inside and U(r)= ∞ outside the well) can be written as
2 H =− ∇2 +U(r) (4.3.1.1) 2m where is Planck’s constant, m is the mass of the particle (an electron or a hole), ∇ is the Laplacian operator written in spherical coordinates. The resultant energy eigenvalues are given by
22ζ E = nl (4.3.1.2) nl 2mR2
where R is the radius of the QD and ζ nl denotes the n-th root of the l-th order
spherical Bessel function, j(lnlζ ) = 0 . The corresponding wave functions are
61 ⎧ 2 j(lnlζ r/R) m ⎪ 3 Y(l θ,φ ), r≤ R Ψnlm ()r,θ,φ = ⎨ Rj(lnl+1 ζ ) (4.3.1.3) ⎪ ⎩ 0, r>R
m where Y(l θ,φ ) are the normalized spherical harmonics. We can see that the energy levels are labeled by the indices n (the principal) and l (the angular quantum number) and the levels are 21l + degenerate with respect to the m
(magnetic quantum number), because of the spherical symmetry of the problem.
So far we have not mentioned how the electron and hole masses me and mh ,
used in the description above are related to the free electron mass m0 . The answer to this is provided by the general theory of the behavior of a particle in a periodic potential. Briefly, an electron (or a hole) in the presence of a crystal lattice behaves much like a free particle, carries a so-called quasi-momentum k , and with respect to an applied field it behaves as a particle with a mass, determined by the dispersion relation. Near a band extremum where the particle energy can be expanded in power series with respect to the wave vector, omitting terms of order higher than k 2 we obtain the electron/hole effective mass
2 e,h −1 1 dE me,h = 22 (4.3.1.4) dk k=0 for an anisotropic lattice. In general the effective masses are a tensor quantity due to the anisotropy of the lattice. Although the quadratic approximation for the dependence of the band energy on the momentum typically applies only near the band minima/maxima, we will use the simple approximation for the entire band
62 with a constant effective mass, which, thus, is called the parabolic band model.
In the discussion above we have treated the electron and hole as independent particles confined in the same potential well. Numerous experimental measurements indicate that the electron and hole do interact and can even form bound states and this ‘new’ compound particle is known as an exciton.
Therefore in our attempt to find the exciton spectrum we need to consider the
Coulomb interaction between the electron and the hole. This problem is almost identical to the case of the hydrogen atom, the difference being that instead of the much heavier proton acting as a positively charged particle, we have the hole.
We will give only the final results, ignoring the presence of the boundary for the time being. We know that the two-body problem can be solved by introducing a
Hamiltonian with a new set of coordinates and masses:
222e H = −∇−∇−22 (4.3.1.5) 22M ρ μ r r
where M =+mmeh is the sum of the electron and hole masses,
ρ =+(meerr m hh )/(m e + m h ) is the radius vector of the center of mass,
μ =+mmeh /(m e m h ) is the reduced mass, r is the relative coordinate, and e is the elementary charge. The resultant spectrum consists of continuous energy related to the motion of the center of mass and the quantized ‘internal’ states, given by
63 e22 E,nb=−22 a = (4.3.1.6) 2anb μe
where ab is the well-known Bohr radius, equal to 0.529 angstroms in the hydrogen atom, and n is the principal quantum number.
The above result allows us to treat the electron-hole pair in a QD as a two-particle system, subject to Coulomb interaction which leads to the possibility of formation of bound states. In the presence of a crystal lattice the interaction between the carriers is screened and the exciton Bohr radius is modified by the
22 dielectric constant ε as ab = ε /( μe). The large value of the dielectric constant in semiconductors combined with the small value of the reduced mass results in an exciton Bohr radius substantially larger than the one in the hydrogen atom. We can now rewrite the exciton dispersion relation as
eK222 E(K)ng=− E 2 + (4.3.1.7) 22εanb M
where Eg is the bulk material band gap, K is the exciton wave vector and the last term is associated with the free center-of-mass motion.
The relation between the radius of the quantum dot and the exciton Bohr radius has important implications on the exciton energy spectrum. In the case
when the quantum dot radius R is much larger than the exciton Bohr radius ab the finite size effects are manifested as quantization of the center-of-mass motion.
This regime is known as the weak confinement regime and the exciton energy is given by
64 2 22 e ζ jl EEnjl=− g 22 + (4.3.1.8) 22εanb MR
On the other hand in the strong confinement regime, when R ab , the electron and the hole do not possess a bound state and their motion is largely uncorrelated.
As a result they are treated as independent particles with energies
22ζζ 22 E,eh==−−nl EE nl (4.3.1.9) nl22 nl g 22mReh mR
The Coulomb energy between them can be treated as a perturbation [126]. The
intermediate confinement regime is defined when Ra≈ b and no simple analytical solutions exist.
In summary, we have presented a simple treatment of the effect of quantum confinement on the position of the electron and hole energy levels. The model essentially treats the carriers in a ‘particle-in-a-box’ picture. It assumes that the masses of the carriers are the effective masses inferred from bulk measurements, and identifies different regimes based on the relation between the QD radius and the exciton Bohr radius. In the parabolic band model the only parameters that distinguish among different materials is the carrier effective masses (and the dielectric constant). This apparently is an oversimplification of the problem, and in the next two Sections we will present more advanced multiband calculations of the electronic structure of CdSe and PbSe quantum dots. Since most experiment conducted in this thesis concerns QDs in the regime of strong quantum confinement, we will focus on the theoretical treatment of this regime in
65 what follows.
4.3.2 Multiband model for CdSe QDs
CdSe is a II-VI compound semiconductor that crystallizes either in the zinc-blende or in the wurzite structure, and has a direct band gap of 1.84 eV located at the center of the Brillouin zone. It shares a common band structure in the vicinity of the band gap with other semiconductors such as GaAs, InAs, CdTe,
InSb, and etc. It has a doubly (spin) degenerate conduction band, a four-fold degenerate valence band at the Γ point (‘heavy’ and ‘light’ holes), and a doubly degenerate spin-orbit split-off valence band. One of the most fruitful methods for calculating the band structure in such compounds has been the kp⋅ method.
Since a correct estimation of the electronic structure of the QD involves knowledge about the corresponding dispersion relations in the bulk material, together with the boundary conditions, we will review some basic results regarding band structure calculations in bulk materials with structural properties similar to that of CdSe using the kp⋅ method, and then show how the presence of a surface leads to size-dependent quantized electron-hole levels in the regime of strong quantum confinement.
The behavior of the carriers in the proximity of the band gap in compounds such as CdSe is governed by the interaction of (at least) the above mentioned 8 bands: 2 conduction and 6 valence bands. The full 8-band problem has been
66 successfully implemented in the study of bulk materials [127, 128], as well as in the study of various QDs. It has described the position and size-variation of the exciton levels with excellent accuracy [129, 130]. Inclusion of as many bands as possible improves the quality of the calculation, but lacks simple and/or analytical solutions for intuitive interpretation. As we describe later, most measurements in this thesis utilize the THz radiation with photon energies much less than both the inter- and intra-band transitions. A couple of simplifications can be applied in the model. First, since the heavy hole masses are much larger than the conduction mass, we can ignore the coupling between the conduction and valence bands and teat them separately [131, 132]. This approximation has shown to describe satisfactorily detailed studies on CdSe QDs using the techniques of transient differential absorption and photoluminescence excitation [133, 134]. In addition, the split-off band that arises due to spin-orbit interaction is removed by 0.42 eV (in
CdSe) from the top of the valence band, allowing us to neglect its contribution to the properties of interest described later in this thesis. Therefore instead of dealing with the full 8-band problem, for the time being we shall concentrate on the 4-band problem [135, 136] for the valence electrons (‘heavy’ and ‘light’ holes) and an independent 2-band problem for the conduction electrons, which has been treated in the simple parabolic band model in the previous Section.
The Hamiltonian describing the 4-band system has been developed by
Luttinger and Kohn and has the following form [137, 138]
67 2 ⎛⎞5γγ22k 22 22 22 H =+⎜⎟γ1 −(kJkJkJ)xx + yy + zz 22mm ⎝⎠ (4.3.2.1) 2γ3 −++ ({}{}{}{} kxy k J x J y k yz k J y J z{}{} k xz k J xz J ) m
's 's where γi are the so-called Luttinger parameters, Ji represent the 44× angular momentum matrices corresponding to angular momentum 32/ , and
{kkij} =+ (kk ij k ji k )/2 . This Hamiltonian reflects the symmetry of the crystal lattice of diamond (cubic), and in order to be used in wurzite structures such as
CdSe an additional anisotropic term should be included [136], but in real calculations it is usually neglected. A widely used simplification due to
Baldereschi and Lipari [139] reduces the expression above to the so-called spherical Hamiltonian, which has been shown to be an excellent approximation, leaving the following form
γμ⎡ ⎤ H =−1 pPJ222()()i () (4.3.2.2) 29m ⎣⎢ ⎦⎥
()2 ()2 where p is the carrier momentum operator, μγ= 2 21/ γ , and P and J are spherical tensors of second order, built from bilinear combinations involving the linear and angular momentum operators [139]. The advantage of this
Hamiltonian is that terms responsible for the cubic symmetry have been neglected and the remaining Hamiltonian has spherical symmetry and can be solved analytically. The total angular momentum F = LJ+ is a conserved quantity and can be used to label the energy states, in analogy to the L − S coupling adopted in atomic calculations. The role of spin-orbit coupling in atoms is replaced by the
68 coupling between the envelope and Bloch functions. The form of the Bloch functions follows from the fact that the valence bands arise from degenerate p-orbitals, which when coupled to the spin result in the doubly degenerate
spin-orbit split-off band, labeled by J12/ , and the 4-fold degenerate ‘heavy’ and
‘light’ bands, labeled by J32/ , in which we are interested in. The Bloch functions
in the notation of J,Jz are given by as [140]
i 32/,±= 32 /∓ (XiY) ±± , 2 (4.3.2.3) i 32/,±=⎡±± 12 /⎣ 2 Z∓ (XiY)∓ ⎤⎦ 6 where ± indicates the projection of the spin component. It can be shown that the second term in Eq. (4.3.2.2) can only couple states for which ΔL =±02, .
This, combined with the spherical symmetry of the problem, allows us to look for solutions in the following form [140, 141] rnLFF= K32 / LJ FFR(F) (r)Y r 32 / J (4.3.2.4) zzzzn,KK,Lz∑∑( ) z KL,LL,J=+2 zz where n is the number of the level of corresponding symmetry, ()K32/LJFFzz z are the Clebcsh-Gordan coefficients, K indicates the two possible values ( L
(F) and L + 2 ) of the orbital angular momentum, Rn,K (r) are radial functions, which will be determined from the Schrodinger equation, taking into account the boundary conditions and normalization, Y are the spherical harmonics, and K ,Lz r/J32z are the Bloch functions of the valence band with band-edge angular
momentum J/= 32. By convention, hole energy states are denoted as nQF ,
69 where Q is the smaller value of L (the other is L + 2 ) and F is the total angular momentum. Using this ansatz for the Schrodinger equation with the
Hamiltonian given by Eq. (4.3.2.2) we get sets of differential equations for the unknown radial wave functions. For F/= 12 we get uncoupled radial equations for L =12, states with the spherical Bessel functions as the solutions of these
equations. The corresponding P12/ and D12/ states are determined by the zeroes of the spherical Bessel functions. The energy levels are determined by
the light-hole effective mass m=m/(lhγ 1+ 2γ 2 ). For F/≥ 32 the energy levels
are determined by the heavy-hole mass m=m/(hhγ 1- 2γ 2 ) and the radial functions are solutions to the coupled differential equations [140]
⎛ ⎛⎞⎛∂∂222122531L( L + ) mE ∂ L +∂++ ( L )( L ) ⎞⎞ ⎜ −+(C)1 12 + − − C + + ⎟ ⎜⎟⎜2222 2 ⎟⎛⎞(F) ⎜ ⎝⎠⎝∂∂rrrr γ1 ∂rrr ∂ r ⎠⎟ Rn,L ⎜⎟= 0 ⎜ ⎛⎞⎛⎞∂+∂+2 21L L(L 2 ) ∂∂++2 2 (L 3 )(L 2 ) 2 mE⎟⎜⎟R(F) ⎜C(C)−+ −++−1 −⎟⎝⎠n,L+2 ⎜ 2 ⎜⎟⎜⎟223 222⎟ ⎝ ⎝⎠⎝⎠∂∂rrrr ∂∂ rrrr γ1 ⎠
(4.3.2.5) where the C-coefficients are related to the Wigner 6 − j symbols. The form of the coupled equations suggests solutions in the form of combination of spherical
Bessel functions, which in the case of infinite confinement potential can be taken as [135, 140]
70 ⎛⎞j( βkr/ R)j (k) R(r)Aj(kr/R)(F) =−LL n,L⎜⎟ L ⎝⎠j(L βk) ⎛⎞j(βkr / R ) j (k) R(r)A(C/C(C/C))j(kr/R)(F) =− +2 +1 − LL++22 n,L++212122⎜⎟ L ⎝⎠j(L+2 βk)
(4.3.2.6)
where A is a normalization constant, β = m/mlh hh and k satisfies the transcendental equation
j( βk)j (k)(C/C++ (C/C)2 1 ) LL+21212 (4.3.2.7) 2 =−+j(LL+21212βk)j (k)(C/C (C/C)1 ).
In summary, we have presented a 4-band effective mass model for the calculation of the quantized hole energy levels and the corresponding wave functions. The model correctly incorporates the crystal structure of the sample and accounts for the interaction of the top 4 originally degenerate valence bands, neglecting the effect from the neighboring spin-orbit band and the conduction band. The main result is that the presence of the boundary conditions leads to quantization of the energy levels and admixture of different bands.
4.3.3 Multiband model for PbSe QDs
PbSe is a IV-VI compound semiconductor with direct band gap of 0.28 eV located at 4 equivalent L points in the Brillouin zone. Along with the other lead salts, PbS and PbTe, its crystal structure has the symmetry of the NaCl lattice, and all compounds show qualitatively similar band structures. A unique feature
71 of PbSe is the smallness of both the conduction and valence band effective masses, resulting in Bohr radii for both carriers much larger than the QD radius, leaving both the electron and hole in the strong confinement regime. Dimmock and Wright made the first attempt to use the kp⋅ formalism to study the lead salts [142]. They concluded that there are 6 bands, 3 valence and 3 conduction bands in the vicinity of the band gap that need to be included in the calculation of the electronic structure. Cyclotron resonance experiments [143] in n − and
p − doped lead salts indicated that the conduction and the valence electrons have similar effective masses, and that of the original 6 bands only 4 are located close to the band gap and interact appreciably. This fact illustrates a major difference in comparison to CdSe and materials alike: the smallness of the energy gap and the nearly identical electron and hole masses require a simultaneous treatment of the conduction and valence bands, unlike the case of CdSe where, for simplicity, we could ignore the coupling between the far-lying conduction band and the valence bands. The symmetry of the crystal lattice combined with the spin-orbit interaction determines the form of the Bloch functions for lead salts at the L point to be [144]:
++ ++ L61(L )↑ =− i R ↑ , L(L 61 ) ↓ = i R ↓ , (4.3.3.1) −− −− L(L62 )↑= Z ↑ , L(L 62 ) ↓= Z ↓ where R is a function invariant under the crystal lattice symmetry operations at the L point, Z transforms as the z-coordinate in the (111) direction, and
72 ↑↓, indicate state of spin up and spin down. This allows us to write the
Hamiltonian as [142, 145]
E22 22 P( k− ik ) ⎛ g kPktlz kz tx y ⎞ ⎜ ++−− 0 ⎟ ⎜ 22mmtl 2 m m ⎟ ⎜ E22 22 P( k+ ik ) ⎟ gtxy kPktlz kz ⎜ 0 ++−− − ⎟ ⎜ 22mmtl 2 m m ⎟ H = ⎜ 22 22 ⎟ Pk P(k− ik ) E k k ⎜ lztx y g t z ⎟ −+++ + 0 ⎜ mm22 mmtl 2 ⎟ ⎜ ⎟ P(k+ ik ) Pk E 22 k 22k ⎜ tx y −−++lz0 g t z ⎟ ⎜ ++⎟ ⎝ mm22 mmtl 2⎠ (4.3.3.2)
222 where Eg is the band gap, kkktxy= + , Pt and Pl are transverse and
+,− longitudinal momentum-matrix elements, and mt,l are transverse and longitudinal effective masses [145]. Due to the significant interaction between the conduction and valence bands the energy dispersion relation is highly non-parabolic and anisotropic. Similar to the case of CdSe it is possible to treat the anisotropic part of the Hamiltonian as a perturbation, leaving us with a spherically symmetric Hamiltonian
22 ⎛⎞⎛⎞Eg kP ⎜⎟⎜⎟+⋅− 1kσ ⎜⎟⎝⎠22mm Hk()= (4.3.3.3) 0 ⎜⎟22 Pk⎛⎞E ⎜⎟k1⋅−+σ g ⎜⎟⎜⎟+ ⎝⎠mm⎝⎠22 where 1 is the 22× identity matrix, σ are Pauli spin matrices, P , m− and m+ are defined through the relations
73 222 ± ±± 32PPPm=+tl ,3/2/1/ = m t + m l (4.3.3.4)
The total angular momentum and the parity are ‘good’ quantum numbers, so we look for solution in the form of
Ψ(rr )=↑+↓+↑+↓ F( )L−−++ F( r )L F( r )L F( r )L 16 26 36 46 (4.3.3.5) F (rrrrr )= {} F1234 ( ),F ( ),F ( ),F ( ) where
⎛⎞⎛⎞lm++12 /⎛⎞⎛⎞ lm −+ 32 / ⎜⎟⎜⎟YYm/−−12⎜⎟⎜⎟ m/ 12 21ll++l 23 l+1 ⎜⎟if ( r )⎜⎟⎜⎟ ig ( r )⎜⎟ ⎜⎟ll⎜⎟⎜⎟+1 ⎜⎟ lm−+12 /m/+12 lm ++ 32 / m/+12 ⎜⎟⎜⎟Yl ⎜⎟⎜⎟− Yl+1 ⎜⎟⎝⎠21ll+ ⎜⎟⎝⎠ 2+ 3 FF π ,j,m()rr==⎜⎟ π ,j,m () ⎜⎟ ⎛⎞lm−+32 / ⎛⎞lm++12 / ⎜⎟⎜⎟Y m/−12 ⎜⎟⎜⎟Y m/−12 ⎜⎟23l + l+1 ⎜⎟21l + l f(r)⎜⎟g(r)⎜⎟ ⎜⎟l+1 ⎜⎟⎜⎟l ⎜⎟ ⎜⎟lm++32 / m/+12 ⎜⎟lm−+12 / m/+12 ⎜⎟⎜⎟− Yl+1 ⎜⎟⎜⎟Yl ⎝⎠⎝⎠23l + ⎝⎠⎝⎠21l + jl=+12 /,π =−( 1 )l+1 j = ( l + 1 ) − 12 / ,π =−() 1l
(4.3.3.6)
Here the 2 vectors differ in parity π , j denotes the total angular momentum and m is its projection along the z-axis, and f ’s and g ’s are unknown radial functions. Using this ansatz in the Schrodinger equation results in sets of ordinary differential equations for the unknown radial functions. The form of these equations suggests solutions in the form of linear combinations of spherical
jl and modified spherical il Bessel functions, and a dispersion relation in the form 1 E(k)=±++γk(E22222αk) β k (4.3.3.7) ± 2 ( g ) where ± refer to the conduction and valence bands and
74 22⎛⎞⎛⎞11 11 2 P γ =−=+=⎜⎟⎜⎟−+, α −+, β (4.3.3.8) 22⎝⎠⎝⎠mm mm m
The boundary condition under which the radial functions are subjected leads to the following transcendental equations which determine the possible eigenvalues
ρ±+(k)jll11 (kR)i(λ ±R)− μ ±(k)j(kR)i ll +± (λ R)= 0 π =−()1 l+1 (4.3.3.9) ρ±+±±+±(k)j(kR)ill11 (λ R)+ μ (k)j l (kR)i( lλ R)= 0 π =−()1 l where
2 ρ±±(k)=++−() Eg (αγ)k2 E ( k ) / βk 2 μ±±±±(k)=++() Eg (αγ)λ (k) −2 E (k) / βλ (k) (4.3.3.10) 24αE ++β 2222( αγ −)k +γE (k) λ (k)= g ± ± αγ22−
Having established the possible values of the wave vectors k and λ()k we find the radial functions up to a normalization constant as
j(kR)l f(r)ll=− j(kr) i( lλr) i(l λr)
⎡ j(kR)l+1 ⎤ f(r)ll++11=−ρ(k)⎢ j (kr) i l + 1 (λr)⎥ ⎣ i(l+1 λr) ⎦ (4.3.3.11) j(kR)l+1 g(r)j(kr)ll++11=− i( l + 1λr) i(l+1 λr)
⎡⎤j(kR)l g(r)lll=−ρ(k)⎢⎥ j(kr) − i(λr) ⎣⎦i(l λr)
The model has been used successfully to explain the main, but not all, features appearing in the linear absorption spectra of PbSe QDs [146]. In spite the fact that inclusion of the anisotropic part originally showed that only a small correction to the energy levels should occur [145], a more thorough treatment
75 [147, 148] has shown that the presence of the anisotropic part can lead to the appearance of additional energy levels and a modification of the possible optical transitions, giving an explanation to the presence of the second peak in the spectra, that was not accounted for by the model we treated so far.
Thus, we have described a 4-band effective mass model for PbSe QDs and presented the analytical results for the quantized energy levels and their corresponding wave functions for electrons and holes. The model includes the interaction between the two valence and the two conduction bands and accounts for the specific symmetry properties imposed by the rock salt crystal structure.
The main feature of this model is the significant nonparabolicity of both the conduction and valence bands.
4.4 Limitations on the use of the kp⋅ method
The kp⋅ method has emerged as the ‘standard model’ for describing the electronic structure of QDs [149]. The model is especially appealing because in many cases it allows for analytic solutions and has been successfully used to explain the band structure in various systems of QDs [134, 145]. The method relies on expanding the carrier wave function inside the QD using a small number of band-edge Bloch functions. In reality though, the use of a limited set of Bloch functions leads to an error in the determination of the carriers’ effective masses, which increases when the set of Bloch functions is decreased [150]. It has been
76 shown [151] that the inaccuracy becomes even more pronounced with the reduction of the dimensionality of the investigated object. A thorough investigation of this problem by Fu et al. [150] has shown that the kp⋅ method can be significantly improved by enlarging the set of Bloch functions and introducing the appropriate coupling between the conduction and valence bands.
Doing so, though, makes the problem of finding the electronic structure untraceable analytically.
A recent investigation by An et al. [152], has shown that ignoring the intervalley coupling and reducing the number of interacting bands in the vicinity of the band edge in the kp⋅ model has profound effects on the electronic configuration in PbSe QDs. A sophisticated pseudopotential calculation incorporating all these effects has produced a much different band structure.
The origin of the experimentally observed exciton absorption peaks is naturally identified, and the valence states are found to be much more densely spaced than the conduction states, invalidating the picture of similar electron and hole energy levels.
These examples indicate that a direct application of the kp⋅ model to a particular system of QDs may lead to a misleading electronic band structure, and should be done after a careful consideration of all material specifics.
77 Chapter 5
Terahertz Electric Polarizability of Excitons in PbSe and CdSe
Quantum Dots
The ability to manipulate charge carriers in nanostructures is essential to the fundamental studies of the behavior of quantum-confined carriers and the applications of these materials in electronics and optoelectronics [86]. External electric fields are often applied for this purpose. Therefore, characterization of the response of charge carriers confined on the nanometer length scale to externally applied electric fields is of both fundamental and technological importance. Isolated monodisperse colloidal nanoparticles of controlled size provide an ideal laboratory for a systematic study of quantum-confined charge carriers. The electric response of an exciton (electron-hole pair) quantum-confined in QDs, parameterized as exciton polarizability, has been inferred from Stark shift measurements for CdSe QDs [153-155]. Direct characterization of exciton polarizability, however, is difficult since it requires electrical contacts to the QDs and in addition, excitons in QDs generally live for a very short period of time. Recently, Wang et al. [56] demonstrated such a measurement using the method of terahertz time-domain spectroscopy
(THz-TDS), in conjunction with an ultrafast optical excitation. The technique
78 allows the direct characterization of the exciton polarizability with a freely propagating THz pulse (thus lifting the requirement on the contacts) and to do so with picosecond time resolution [9]. These capabilities have also been exploited in recent studies on charge transport and carrier dynamics in bulk solids, [48, 55,
59, 156-159] quantum wells, [160-162] and QDs [56, 163-165]. It has been shown [56] that the response of excitons strongly quantum-confined in CdSe QDs is ‘atom’-like, characterized by a large polarizability with an instantaneous response up to THz frequencies. Although it is known that the exciton polarizability scales as the forth power of the QD radius [56] and can even be larger in elongated QDs or nanorods, [166] its dependence on the QD material, crucial information for the development of nanostructures with large electric response, yet remains unexplored.
In this Chapter we apply the technique of THz-TDS to reveal the dependence of exciton polarizability on the material parameters through a systematic comparison of QDs made of PbSe and CdSe. PbSe and CdSe, normally exhibiting in colloidal QDs rocksalt [167] and wurzite [90] lattice symmetry, have drastically different electronic band structures [168]. These materials have also attracted significant attention recently for potential applications in lasers, light-emitting diodes, and photovoltaic devices [86].
79 5.1 Response of a single exciton to THz radiation
In the regime of strong quantum confinement with QD radius R < exciton
Bohr radius, as investigated here, the electron and hole comprising an exciton are largely uncorrelated [130]. We can, thus, analyze the exciton approximately within a single-particle picture and treat its polarizability as the sum of independent contributions from the electron and hole. The dc polarizability of the electron can be estimated following α ~er 2 /ΔE , [169] where e is the elementary charge, er the transition dipole moment, and ΔE the energy level spacing. A typical value for the dipole moment is ~ eR and for the level spacing
22 4 22 is ~ /( me R ) , which results in a polarizability ~ R /ae , with a/(me)ee=
denoting the electron Bohr radius. Here me is the electron effective mass and
is Planck’s constant. The hole contribution can be estimated similarly. It is valid to approximate the low-frequency polarizability of an exciton at frequency ω , relevant for the THz measurements, by its dc value if ΔE >> ω . For THz frequencies (1 THz ≈ 4 meV) and carrier effective mass ~ the electron mass m , it requires R <10 nm, which is satisfied for most semiconductor QDs in the regime of strong quantum confinement. The simple dimensional analysis reveals the correct size dependence for the exciton polarizability (∝ R4 ). It also indicates that the polarizability increases linearly with the carrier effective mass.
We use the carrier band mass to evaluate the effect. For the two materials of
80 interest the relevant band masses (averaged over orientations) are: mh (heavy
hole) = 119.m, m.me = 011 (CdSe) [140] and mmhe≈ ≈ 005 .m (PbSe) [168].
These values suggest that the exciton polarizability is dominated by the hole contribution in CdSe and is roughly equally contributed by the electron and hole in
PbSe QDs. In addition, the excitons are expected to be more polarizable in
CdSe than in PbSe QDs of similar size.
5.2 Experimental setup
The optical pump/THz probe setup was similar to the one described in
Chapter 1. The fundamental and second harmonic of the laser generated in a
0.5-mm-thick BBO crystal were used to inject excitons in PbSe and CdSe QDs, respectively. CdSe/ZnS core/shell nanocrystals (NN-Labs) dissolved in
2,2,4,4,6,8,8-heptamethylnonane (Aldrich) and PbSe QDs (Evidot) dissolved in n-hexane (Aldrich) were used in this study. The QD suspensions were dilute with a volume fraction not exceeding 10-4. The dot radius ranged from 1.2 - 3.3 nm (CdSe) and 2 - 3.5 nm (PbSe) with a typical ±5% size distribution. The size, size distribution, and concentration of CdSe and PbSe [148, 170] QDs were inferred from the linear absorption spectra. Sample thickness of 2-10 mm was typically used and all measurements were performed with samples at room temperature. A control experiment with QD suspensions and neat solvents was conducted carefully to exclude possible contributions from solvents to the
81 pump-induced response. The pump beam size was kept at least 2 times of the probe beam size within the entire sample volume to minimize the effect of pump inhomogeneities [171]. In addition, relatively weak excitation was used to avoid significant multiple exciton effects. Pump fluencies in the range of 0.03 – 0.1 mJ/cm2 for CdSe and 0.15 – 1.5 mJ/cm2 for PbSe QDs, respectively, were typically used, within which the pump induced response was largely linear in pump fluence.
5.3 Results and discussion
Typical experimental data is shown in Fig. (5.3.1(a)): E(t) (solid black line) is the electric-field waveform of a THz pulse transmitted through an unexcited QD suspension (in this case 3.1-nm-radius PbSe QDs); and ΔE(t) (solid gray line) is the change in the transmitted THz waveform due to photoexcitation. The latter was typically recorded approximately 6 ps after photoexcitation to allow excitons to fully relax to the ground state [172]. A typical dependence of the photoinduced response on the pump-probe delay time is shown in the inset of Fig. (5.3.1(a)).
In the regime of weak excitation, the frequency-dependent dielectric response
Δε( ω )induced by photoexcitation in the QD suspension can be extracted through the Fourier transforms of the waveforms as
ΔE( ω ) ωL Δε( ω ) = i (5.3.1) E( ω )c2 ε( ω )
82 where ε( ω ) ≈ ε is the dielectric function of the unexcited QD suspension and L is the thickness of the suspension.
-3 6x10 -7 6x10 -5 (a) 1.2x10 (b) 4 1.0 De'
(a.u.) 2
4 ETHz
THz 0
DETHz x 5000 E 0.8 Δ 0 4 8 0.6 Time delay (ps) (a.u.) 2 De
THz 0.4 E
0.2 0 De'' 0.0
-0.2 -2 -12 12 1 2 3 4 5x10 0.5 0.6 0.7 0.8 0.9 1.0 1.1x10 Frequency (Hz) Time (s) Fig. 5.3.1 THz electric-field waveform transmitted through an unexcited suspension of
3.1-nm-radius PbSe QDs (black solid line) and the photoinduced change in this waveform (grey
solid line). Inset: dependence of the pump-induced response on the pump-probe delay time in
the first 10 ps. (b) Spectral dependence of the photoinduced complex dielectric response of the
QD suspension (solid grey lines). Dashed lines in (b) represent a model with Δε'const=
and Δε'' = 0 and the dashed line in (a) is the change in the THz electric-field waveform
predicted by the model.
The sample was optically thick and multiple reflections at the interfaces were negligible and not included in Eq. (5.3.1). The pump-induced dielectric response
Δε(ω) extracted from the waveforms of Fig. (5.3.1(a)) is illustrated in Fig. (5.3.1(b))
(solid grey lines). Both the real Δε' and the imaginary part Δε'' are largely frequency independent up to ~ 1 THz and the amplitude of the imaginary part
83 Δε'' is approximately zero. The dashed lines represent a model dielectric response with a spectrally flat real part and a vanishing imaginary part
( Δε'const= and Δε'' = 0 ) both in the frequency domain (Fig. (5.3.1(b))) and in the time domain (Fig. (5.3.1(a))). The agreement between experiment and model is satisfactory. Such a response is characteristic of a system with discrete energy levels (such as an atom) while probed with electromagnetic radiation that is far below the resonances. This ‘atom’-like response first observed in small
CdSe QDs [56] was found in all QDs that we investigated. The result confirms that PbSe and CdSe QDs of radii less than their exciton Bohr radius possess well separated electron and hole energy levels with spacing >> 1 THz ~ 4 meV. We note that the pump-induced response in dilute QD suspensions was generally weak and quantitative characterization of excitons in PbSe QD suspensions was
particularly demanding. The normalized dielectric response ( Δε / ε )/nex by
the exciton density nex in the suspension of PbSe QDs, for instance, was typically ~ 350 times smaller than that in the suspension of CdSe QDs of similar size. The small exciton response from PbSe QDs, as we discuss below, is a combined effect of strong dielectric screening and large transition energies, a direct consequence of small carrier masses.
We now convert the pump-induced dielectric response Δε of the QD suspensions to the polarizability, a fundamental property, of individual excitons.
84
Fig. 5.3.2 Schematics of the electric polarizability of an exciton strongly quantum confined in
a QD of radius R to an externally applied electric field F . Within the parabolic-band
effective mass model as described in the text, in the absence of the field (left), the
ground-state exciton possesses a spherically-symmetric electron and hole distribution
centered at the center of the dot (only the electron distribution shown) which result in a zero
net dipole moment. The externally applied electric field perturbs the exciton wave function
and the charge spatial distribution (right), which leads to a net dipole moment P in the
QD.
As shown in Fig. (5.3.2) we define the exciton polarizability as α = P/F. Here
F is the externally applied electric field and P is the dipole moment of the exciton that one measures when the unexcited QD is embedded in a host with a matched dielectric constant. This definition allows us to make a direct comparison of experiment with a simple quantum mechanical model below, where the dielectric properties of the unexcited QDs are not included. In the limit of dilute QD suspensions the exciton polarizability is related to the measured
85 pump-induced dielectric response of the composite by an effective medium theory through
2 36πεh Δε = nex 2 α (5.3.2) ( εεih+ 2 )
where εi and εh are the dielectric constant of the unexcited inclusion (QD) and the host (solvent). In the analysis we used the bulk value of the low-frequency
dielectric constant for the inclusion ( εi = 95. for CdSe [168] and 210 for PbSe
[168] ). The host dielectric constant εh , which roughly equals the dielectric constant of the suspension ε , was determined from the THz transmission
measurements through unexcited QD suspensions ( εh =189. for CdSe and 2 for
PbSe suspension). The exciton density nex in the suspension was inferred from the number of absorbed pump photons and the excitation volume assuming a unit quantum yield of exciton generation.
The extracted exciton polarizability following Eq. (5.3.2) is summarized in Fig.
(5.3.3). The error bars represent the statistical uncertainties of the measurements. Systematic errors such as the effect of spatial inhomogeneities of the pump [171] are hard to quantify and are not included. Systematic error is, however, expected to influence all the measurements similarly and have small effect on the relative amplitude of the exciton polarizability for various QDs. In light of this the agreement of the current experiment with the previously reported work [56] on CdSe QDs is surprisingly good. The exciton polarizability for both
86 CdSe and PbSe QDs is seen to rise rapidly with increasing dot size, roughly following a R4 dependence (dotted lines).
CdSe 100 5 8 CdSe 7
) PbSe
3 6 5 Parabolic-band, CdSe 4 Parabolic-band, PbSe
3 Multiband, CdSe Multiband, PbSe 2
10 8 7 6 Exciton polarizability (nm polarizability Exciton 5 4
3
1.5 2.0 2.5 3.0 3.5 4.0 Radius (nm)
Fig. 5.3.3 Polarizability of photoinduced excitons strongly quantum-confined in CdSe and PbSe
QDs as a function of the QD radius. Details about the experimental results (symbols) and the
theoretical results based on a parabolic-band effective mass model (dotted lines) and multiband
effective mass models (solid lines) are described in the text.
Values as large as 100 nm3 were observed. In addition, the exciton polarizability in CdSe QDs is generally larger than that in PbSe QDs with a ratio of 1.7 ± 0.2 for dots of equal size. These observations on the exciton polarizability, including both its size-variation and relative magnitude, are described qualitatively by the dimensional analysis provided above. For a more quantitative analysis we turn to the theoretical models described in the previous Chapter.
87 5.4.1 Calculation based on the parabolic band model
As discussed above, in the regime of strong quantum confinement the electron and hole comprising an exciton can be described largely as uncorrelated particles. The Coulomb interaction between the electron and hole can be treated as a perturbation [130]. We found the correction to the final value of the exciton polarizability due to the interaction to be modest (~ 10%) for all the dots investigated here. Therefore, in the discussion below we will ignore the interactions for simplicity. We first consider the parabolic-band effective mass model, introduced in Chapter 4 that allows us to derive an analytical solution for the exciton polarizability. For simplicity, we treat the electron and hole as spinless particles, but the result remains unchanged when spin is included.
Below we show the detailed calculation for the electron contribution to the exciton polarizability. The hole contribution can be evaluated similarly. We write the electron wave function as a product of a cell-periodic Bloch function and an envelope function. The single-particle energy spectrum and the envelope wave functions are well-known [130] and are given by Eq. (4.3.1.3) and (4.3.1.9) , and
remember that ζ nl denotes the n-th root of the l-th order spherical Bessel
function jl . A set of quantum numbers (n,l, m) including the principal, angular, and magnetic quantum number is used to describe each state. States
with energy Enl are m -fold degenerate. The electron polarizability in state
88 nlm can be calculated using the familiar result from perturbation theory
2 ′′ 2 nlm'er⋅ nlm αnlm = 2e'∑ (5.4.1.1) nlm'′′ EEnl′′ − nl
Here ∑′ denotes a sum over all states except nlm and e is the unit polarization vector of the external electric field. With the help of the radial integral
2 R 4ζζR rj(3 ζ r/R)j(ζ r/R)dr= n' l' nl (5.4.1.2) ∫ l' n'l' l nl 222 j(l'++11ζ n'l')j l (ζ nl)(0 ζζ n'l'− nl ) the transition dipole moments in the numerator between states nlm and
n' l' m' can be evaluated as
4ζζR ()eD−1 p(p) nlm′′ ′er⋅= nlmnl′′ nl pnlmnlm′′ ′ , 222∑ ( ζζnl′′ − nl ) p,=±01 (l2121++ )(l' )
()±1 δm' m±1 D(′′ =±±+−δ (l' m)(l m2 ) δ (l'∓∓ m)(l m)), (5.4.1.3) nlm'nlm2 l'l+−11 l'l ()02222 Dnlm'nlm′′ =−−−δ m'm( δ l'l+−11l' m δ l'l lm),
where e(i)/± =±12 and e0 =1 are the spherical components of the unit polarization vector e . Combining Eq. (5.4.1.2) and (5.4.1.3), we rewrite Eq.
(5.4.1.1)
64R4 1 ζζ22 α =+'(l'nl′′ nl δ lδ ) (5.4.1.4) nl∑ 225 l',l+−11 l',l 321al(enlnlnl′′ +−ζζ′′ )
22 with the electron (hole) Bohr radius a/(me)e( h )= e( h ) . In Eq. (5.4.1.4) we have
averaged the exciton polarizability αnlm over the m -fold degenerate states,
which is equivalent to averaging αnlm over all possible polarization states of the external field. The electron polarizability is entirely determined by the dot radius,
89 the electron Bohr radius, and a constant determined by the roots of the Bessel functions. For the ground-state exciton, we obtain its polarizability straightforwardly
4 11 αα=+≈eh α 0.0363R ( + ) (5.4.1.5) aaeh
5.4.2 Calculations based on the multiband models
As mentioned earlier for CdSe, the parabolic-band effective mass model adequately describes the electron states because the lowest conduction band is well separated from the remaining bands and non-parabolicity is small. This simple picture, however, breaks down for the valence bands. Because of the strong spin-orbit interaction, the three highest valence bands mix. Since we are interested in the ground-state exciton, we ignored the split-off band (~ 0.42 eV lower than the heavy-hole and light-hole band) in the calculation and used the
4-band (doubly spin degenerate heavy- and light-hole band) effective mass model, presented in Chapter 4. The spherically symmetric kp⋅ Hamiltonian was used to evaluate the hole polarizability since analytical solutions are known [140]. The polarizability in this case can be calculated with the help of
⎛⎞F'1 F F'−+ F'z 32 / + F n'L'F'F'zz Ercos(θ )nLFF= (−++×12121 ) E(F')(F)⎜⎟ ⎝⎠−F'zz0 F ⎧⎫⎛⎞F' F11 K' K (K')(K)R2121++(F') (r)R(r)rdr (F) 3 ∑ ⎨⎬⎜⎟ ∫ n',K' n,K K,K'⎩⎭⎝⎠KK'32 / 000
(5.4.2.1)
90 where we have used the properties of the Clebsch-Gordan coefficients and the
Wigner 6-j symbols. Our calculation showed that the main contribution to the polarizability arises from transitions that involve only the first few heavy-hole states. Therefore, the parabolic-band and multiband effective mass model predict similar exciton polarizabilities for CdSe QDs when the heavy hole mass is used in the parabolic band model.
PbSe, on the other hand, is a narrow gap semiconductor. Interactions between the lowest conduction and the highest valence band are expected to be strong, leading to pronounced band non-parabolicity and breakdown of the simple parabolic-band model. Similarly as for CdSe, analytical solutions can be derived for a 4-band (doubly spin degenerate conduction and valence band) effective mass model under the spherically symmetric approximation for the kp⋅
Hamiltonian [145]. These solutions were used to evaluate the exciton polarizability. The significant conduction-valence band mixing reduces the level spacing of dipole-allowed transitions and increases the magnitude of the exciton polarizability. For instance, for QDs of 3-nm radius inclusion of the conduction-valence band mixing reduces the first transition energy from about 0.7 eV to 0.3 eV for both carriers.
Calculation based on parabolic band model with the earlier given carrier
masses ( m.mh =119 , m.me = 011 for CdSe and mmhe≈ ≈ 005 .m for PbSe) is included in Figure 5.3.3 as dotted lines. It is clear that the smaller carrier
91 masses in PbSe result in a larger electron and hole Bohr radius, and consequently, a smaller exciton polarizability. The parabolic-band effective mass model describes very well the measured exciton polarizability for CdSe QDs, but fails to do so for PbSe QDs. It underestimates the magnitude of the response by about
4 times. The 4-band model (green solid line) is seen to describe satisfactorily the experimentally measured exciton polarizability in PbSe QDs. Although a recent atomistic pseudopotential calculation [152] has found a more subtle electronic structure for PbSe QDs than what kp⋅ calculations have suggested, the ground-state exciton polarizability does not seem to be influenced significantly, likely because it is determined primarily by transitions involving the few lowest states.
5.5 Conclusions
The THz electric response of excitons strongly quantum-confined in QDs was found to be ‘atom’-like and vary roughly as the dot radius to the fourth power, in excellent agreement with the recently reported results on CdSe QDs [56]. A systematic comparison of QDs made of PbSe and CdSe revealed that the exciton polarizability rises with the increase of the carrier effective mass. Excitons in
CdSe were found to be about two times more polarizable than in PbSe QDs of similar size if they were probed in media with dielectric constants matching that of the unexcited QDs. However, since semiconductor QDs often have much larger
92 dielectric constants than that of the host medium, the observable exciton response is smaller due to dielectric screening of both the external electric field and the induced dipole moment. These experimental findings were well described by an effective mass model. The simple parabolic-band model was shown to provide the correct size variation and the order of magnitude for the exciton polarizability; the multiband model with realistic electronic band structures for the QD material predicted satisfactorily the magnitude of the exciton polarizability.
93 Chapter 6
Multiexciton Auger Recombination in CdSe Quantum Dots
Studied with THz-TDS
In the previous Chapter we investigated the response of optically generated carriers in semiconductor QDs at a fixed time shortly after photoexcitation. More insight into the behavior of electron-hole pairs in nanocrystals is provided by their dynamics on different time scales. The dominating relaxation channels on the nanosecond scale are the radiative recombination and surface/defect trapping.
On a much shorter scale a non-radiative relaxation channel, known as Auger recombination, is significantly enhanced. It is characterized by an ultrafast
(picosecond) transfer of energy from one electron-hole pair to a third particle [173].
This process, although present in bulk semiconductors as well, becomes the dominant multiparticle relaxation mechanism in QDs due to the spatial confinement and the lack of momentum conservation [87, 174]. Studying and potential controlling of the effect of Auger recombination in various systems of semiconductor QDs [123] has attracted significant attention because of possible applications of nanocrystals regarding optical amplification and lasing [175], applications to photovoltaics [122], generation of entangled states, etc.
94 The small number of electron-hole pairs in photoexcited QDs leads to the possibility of monitoring quantized Auger lifetimes, observed in various material systems [87]. Klimov et al. [173] have shown that the Auger decay rates vary cubically with the size of the nanocrystal and with respect to the carrier density they behave similarly to the bulk material.
Information about the dynamics of multiple excitons is usually obtained by resonantly probing the depopulation of the ground exciton state, a method sensitive to the population dynamics primarily of the electrons [173]. In this
Chapter we present measurements using, alternatively, the non-resonant optical pump/THz probe technique. We note the principal difference of measuring the exciton dynamics using THz-TDS and Transient Absorption bleaching (TA). The significant difference between the electron and hole effective masses results in a hole energy level spectrum that is much more dense than the electron one. As a consequence the resonant method of TA is very sensitive to a change in the number of electrons present in the lowest exciton state, since the holes can be found with finite probability, at room temperature, at various adjacent closely-located levels. In contrast, the very low energy of the THz probe photon,
4 meV, indicates a much larger sensitivity to the behavior of the holes. The non-resonant nature of this experimental technique results in a signal arising from the collective response of all hole carriers populating various energy levels.
95 Our goal in this experiment is to use THz-TDS to observe ultrafast Auger recombination, characterize the relaxation lifetimes and compare this to existing data.
6.1 Experimental setup
The optical pump/THz probe setup was similar to the one described in
Chapter 1 and used in Chapter 5. The second harmonic of the TI:Sapphire laser was used to inject excitons in CdSe QDs. Mono-disperse CdSe NPs stabilized by dendron ligands (NN-labs) in the range of 1.1 – 3.5 nm in radii and with a typical size-distribution of 5% were used for the measurements. They were dissolved in 2,2,4,4,6,8,8-heptamethylnonane (Aldrich) with a typical volume fraction of ~ 10-4. The particle size and concentration were determined from the wavelength of the first absorption peak and the absorbance [170]. Samples in
2-mm quartz cells were optically thin to ensure good excitation uniformity. The average number of excitons injected was varied from 0.1 to 3 per QD. All measurements were done with samples at room temperature.
6.2 Results
A typical time-domain measurement is shown in Fig. (6.2.1): E(t) (solid black line) is the electric-field waveform of a THz pulse transmitted through an unexcited QD suspension (in this case 3.3-nm-radius CdSe QDs); and ΔE(t)
96 (solid red line) is the change in the transmitted THz waveform due to photoexcitation.
-5 2.5x10 (b) Δε' -3 (a) 6x10 2.0 1.5
Δε 1.0 ΕTHz 0.5 ΔΕ x 500 Δε'' THz 0.0 4
12 0.4 0.6 0.8 1.0x10 Frequency (Hz) (a.u.)
THz 2 Ε
0
-2 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Time (ps) Fig. 6.2.1 THz electric-field waveform transmitted through an unexcited suspension and the
photoinduced change in the waveform (red line) for an average of < N.>=15 excitons per
QD. Inset: spectral dependence of the photo-induced complex dielectric response of the QD
suspension (solid lines). Dashed lines in the inset represent a model with Δε'const= and
Δε'' = 0 ; dashed line in the main panel is the change in the THz electric-field waveform
predicted by the model.
In the regime of weak excitation, the frequency-dependent dielectric response
Δε( ω ) induced by photoexcitation in the QD suspension can be extracted
97 through the Fourier transforms of the waveforms with the help of Eq. (5.3.1).
The pump-induced dielectric response Δε(ω) extracted from the waveforms of Fig.
(6.2.1) is illustrated in the inset. Both the real Δε' and the imaginary part Δε'' are largely frequency independent up to ~ 1 THz and the amplitude of the imaginary part Δε'' is approximately zero. The dashed lines represent a model dielectric response with a spectrally flat real part and a vanishing imaginary part
( Δε'const= and Δε'' = 0 ) both in the frequency and in the time domain. Such a response is characteristic of a system with discrete energy levels (such as an atom) while probed with electromagnetic radiation that is far below the resonances. This ‘atom’-like response was found in all QD samples, at all exciton populations and was independent of the delay between the photoexcitation and measurement of the induced response. This allowed us to use the real part of the dielectric constant Δε' to characterize the QD response.
The dynamics of the photoexcited carriers was observed by monitoring a single point on the induced THz waveform (the positive peak) and varying the delay between the optical excitation and THz probe pulse. This corresponds to a mapping of the response of all the excitons present in the system as a function of time. The carrier dynamics at a varying average exciton population for a 3.3-nm nanoparticle is presented in Fig. (6.2.2). The response is seen to change extremely fast in the first few tens of picoseconds, especially at higher < N > , indicating the onset of the Auger recombination channel. After several hundred
98 ps the dynamics can be viewed as a single-exponential decay process, characteristic of a radiative recombination of electron-hole pairs. Qualitatively similar behavior was observed for all other samples.
3.0
1.5
1.0
0.5
0.0 0 20 40 60 80 100 120 140 Time (ps)
Fig. 6.2.2 Pump-dependence of different average exciton populations in a 3.3-nm CdSe QD.
The linear relation between Δε and ΔETHz allows us to express the real part of
Δε as a function of the average exciton population < N > , thus relating the
directly measured dynamics of ΔETHz to the dynamics of <>N . This calibration curve for a 3.3-nm CdSe QD is presented in Fig. (6.2.3). The measurements were done at 3 ps after photoexcitation (see Chapter 7 for details).
99 50
40
30
/C, L/mol 20 De
10
0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Fig. 6.2.3 Dependence of the normalized real part of the induced complex dielectric response
(by the molar concentration) on the average number of excitons per QD (black dots). Red line:
fit to a power law, y = Ax p , p ≈ 05. .
6.3 Discussion
The quantized Auger recombination lifetimes can be extracted by solving the following system of coupled differential equations
dn n dn n n dn n n NNNNN=−, −−11 = −,..., 121 = − (6.3.1) dt τNNNdt ττ−121dt ττ
which describes the multiexciton decay through N finite steps. Here nN is the fraction of nanoparticles containing N electron-hole pairs, a quantity calculated by assuming a Poissonian distribution for a given < N > . The initial conditions
100 for this system are provided by n(tN = 0 ) and the solution is given by a sum of
−t/τN exponential terms: <>=N(t )∑ AN e , where AN are time-independent N coefficients and τN are the lifetimes of the QD state containing N excitons.
Klimov et al. [173] have shown that instead of numerically solving Eq. (6.3.1), a simple subtractive procedure can be adopted to reliably determine the unknown decay rates. This procedure consists of matching the long-term portions
(normalizing) of all dynamics curves, since they are dominated by a singly excited
QDs, and subtracting the trace corresponding to the lowest average exciton number from all other traces. Thus we are left with the dynamics of at least
doubly excited QDs, yielding τ2 . Upon further normalization and subtraction the time-constants of the relaxation of higher number of electron-hole pairs can be determined. It was also shown that the results are not sensitive to the exact initial QD population. We have adopted the same procedure to extract the Auger rates and the results are presented in Fig. (6.3.1). The values for the Auger rates we obtained (red symbols) closely match those reported previously (black symbols). For the range of sizes that we investigated they vary cubically with the radius of the nanoparticles.
101 1000 8 6 4
2
100 8 6 4
2 τ2 τ 10 3 8 τ Relaxation time (ps) time Relaxation 6 4 4 τ2 (Klimov et al.)
τ3 2 τ4 1 2 3 4 5 6 7 8 9 1 10 QD radius (nm)
Fig. 6.3.1 Size-dependence of multiexciton recombination lifetimes obtained via THz-TDS (red
symbols). Black symbols: data obtained via transient absorption spectroscopy. Straight lines:
fits to a R3 dependence.
We also performed independent transient absorption measurements on the same QD samples, which reproduce the results obtained via the optical pump/THz probe technique. Our experiment indicates that both methods measure similar exciton dynamics and result in similar Auger recombination lifetimes. The drawback of THz-TDS comes from the fact that the weak non-resonant response has to be averaged numerous times in order to achieve a signal-to-noise ratio that allows for reliable extraction of the decay rates.
102 6.4 Conclusions
In summary, we have used the method of THz-TDS to observe ultrafast multiexciton recombination due to the Auger effect in CdSe QDs of varying radii.
The extracted relaxation lifetimes scale linearly with the volume of the nanoparticle and their values closely match those obtained with the help of resonant transient absorption bleaching. These results indicate that the non-resonant optical pump/THz probe time-resolved technique can be used to accurately follow the multiexciton dynamics in semiconductor quantum dots.
103 Chapter 7
Response of Multiple Excitons in CdSe Nanoparticles Studied
with Terahertz Time-Domain Spectroscopy
Characterization of the response of charge carriers confined on the nanometer length scale to externally applied electric fields is of both fundamental and technological importance. In Chapter 5 we presented results from a comparison study of the response, parameterized as polarizability, between two different systems of QDs, CdSe and PbSe, using THz-TDS. These investigations were performed in the limit of exciting much less than one electron-hole pair in a QD, precluding many-body effects. It is of both theoretical and experimental interest to study the behavior of the exciton polarizability in the case of multiple charged carriers confined on a nanometer length scale [133, 176-179]. The aim of this study is to account for the effect of Auger recombination, presented in the previous Chapter, and investigate how the response of multiple excitons is modified by the presence of carrier-carrier Coulomb interaction. We use the method of THz-TDS to study the response of multiple excitons to an applied THz electric field in CdSe QDs of varying radii. We perform time-resolved measurements of the electron-hole pairs’ response at a delay time of 3 ps to allow the ensemble of optically generated excitons to relax to its ground state, while
104 recombination is still insignificant. Modeling the polarizability in the language of independent carriers is satisfactory only in the limit of low exciton number, and with increasing carrier density the mutual Coulomb interaction cannot be ignored.
7.1 Experimental setup
The optical pump/THz probe setup was similar to the one described in
Chapter 1. In order to create above-band-gap excitation, approximately 90% of the pulse energy out of the amplifier was employed to generate second harmonic at 3.1 eV in a 0.5-mm-thick BBO crystal. Mono-disperse CdSe NPs stabilized by dendron ligands (NN-labs) in the range of 1.1 – 3.5 nm were used, with a typical size-distribution of 5%. Dilute solutions of CdSe dots (volume fraction ~ 10-4) in
2,2,4,4,6,8,8-heptamethylnonane (Aldrich) were used in the measurements. The particle size and concentration were determined from the wavelength of the first absorption peak and the absorbance at this wavelength [170]. The average number of excitons per QD was in the range 0.1 – 3 for all samples. Sample thickness of 2 and 10 mm was used and all measurements were done with samples at room temperature.
7.2 Results
A typical time-domain measurement is similar to the one shown in Fig. (6.2.1):
E(t) (solid black line) is the electric-field waveform of a THz pulse transmitted
105 through an unexcited QD suspension and ΔE(t) (solid red line) is the change in the transmitted THz waveform due to photoexcitation. The latter was recorded approximately 3 ps after photoexcitation to allow excitons to fully relax to the ground state [172]. Both the real Δε' and the imaginary part Δε'' are largely frequency independent up to ~ 1 THz and the amplitude of the imaginary part
Δε'' is approximately zero. The response is characteristic of a system with discrete energy levels (such as an atom) while probed with electromagnetic radiation that is far below the resonances. This ‘atom’-like response first observed in small CdSe QDs [56] and confirmed by the experiment presented in
Chapter 5, was found in all QDs and at all exciton populations. This allows us to use the real part of the dielectric constant Δε' to characterize the QD response.
Figure (7.2.1) (red dots) represents a typical dependence of Δε' on the average number of excitons per QD for a 2.58 nm nanoparticle. It is seen to increase linearly in the limit of small excitation density, and saturating at larger populations. Qualitatively we observe similar response for all QD samples.
Similarly to the procedure described in Chapter 5 we can convert the pump-induced dielectric response Δε of the QD suspensions to the polarizability, a fundamental property, of the excitons. In the limit of dilute QD suspensions the exciton polarizability is related to the measured pump-induced dielectric response of the composite through Eq. (5.3.2).
106 80 original data data at 3 ps Parabolic band model
60
/C, L/mol 40 De
20
0
0.0 0.5 1.0 1.5 2.0 2.5
Fig. 7.2.1 Dependence of the normalized real part of the induced complex dielectric response
(by the molar concentration) on the average number of excitons per QD (red dots). Black dots:
recalculated exciton population, accounting for the Auger recombination. Solid line: model of
non-interacting carriers.
In contrast to the previously investigated limit of a single exciton per QD, in this case the polarizability is a sum of contributions arising from different number of electron-hole pairs, αα= ∑ N PN . Here PN denotes the probability of finding a N nanoparticle with N excitons, a quantity that can be estimated taking into account, e.g. the absorption spectrum, and assuming a Poisson distribution of photoexcited carriers. As previously discussed the exciton population rapidly decreases due to the Auger recombination, therefore the estimation of the
107 average number of electron-hole pairs at the time of the photoexcitation (time
‘zero’) and at the time of the measurement is different and needs to be accounted for. This analysis is done in the following Section.
7.3 Data analysis
The measurements of the pump induced change in the transmitted THz field were recorded at a 3 ps delay. As previously shown the ultrafast population
dynamics in NPs can be characterized by a discrete set of time constants τN , describing the evolution from N to N −1, etc., excitons due to the non-radiative
Auger recombination. In Chapter 6 we showed that the Auger constants depend cubically on the radius of the quantum dot and depend on the carrier density similarly to bulk material. Thus we can use the decay times previously obtained to find the time constants corresponding to a larger number of electron-hole pairs.
As a rule, for any given average number of excitons we have included, based on the Poisson distribution, sets of QDs contributing by no less than 1%. Thus in our analysis the largest exciton number considered is N = 8 . We assume that the decay of multiexciton states can be described by the set of coupled rate equations Eq, (6.3.1). This linear system can be written in a vector form dn N = M ⋅n (7.3.1) dt N where the matrix M consists of the time constants and the initial conditions are given by the distribution of population at time ‘zero’. Diagonalizing this matrix
108 allows us to write the equation above as dn N = SDS−1 ⋅ n (7.3.2) dt N where the diagonal matrix D contains the Auger time constants and S are the −1 corresponding eigenvectors. Defining a new variable x(t)≡ S nN we can readily solve the resultant system for x(t). Propagating the solution for 3 ps
and reverting back to nN gives as us the redistribution of the population at the required delay due to the non-radiative Auger recombination. This essentially results in a modified average number of excitons per quantum dot which is reflected (black dots) in Fig. (7.2.1). This procedure for calculating the correct
<>N at the time of the measurement was performed for all samples.
7.4 Inclusion of the Coulomb interaction and discussion
In Chapter 5 we obtained an analytical expression for the electron (hole) polarizability for an arbitrary energy level, considering the parabolic-band model.
The degeneracy of each state can be estimated to be 22(l+ 1 ), where l is the orbital angular momentum and 2 accounts for the spin degeneracy. If we ignore any carrier-carrier interactions we can compute the polarizability arising from a given number of excitons with the help of Eq. (5.4.1.4), and then use Eq.
(5.3.2) to convert this to Δε , the readily measured quantity. The result of this calculation is presented (green line) in Fig. (7.2.1). We can see that this non-interacting approximation predicts closely the QD response only in the limit of
109 low average number of excitons and significantly overestimates the results as this number increases. This trend holds true for all measured samples, implying that ignoring the mutual carrier-carrier interaction is an oversimplification and inclusion of many-body effects is necessary.
-21 250x10
Parabolic model 200 Effective mass model
3
150
100 Polarizability, cm Polarizability,
50
0 0 2 4 6 8 10 Number of electrons N
Fig. 7.4.1 Dependence of the hole polarizability on the number of carriers calculated with the
parabolic-band and the multiband effective mass models for a 2.58-nm QD. In both models
the conduction band is treated with the help of the parabolic-band model.
We note that an estimation of the polarizability in CdSe QDs following the multiband effective mass model gives results very similar to the much simpler parabolic-band model (see discussion in Chapter 5). As shown in Fig. (7.4.1)
110 this holds true for up to 6 excitons. This fact permits us to estimate the correction to the polarizability due to the many-body effects utilizing the energy spectrum and wave functions provided by the parabolic-band model.
A complete treatment of the problem of multiexciton polarizability in a QD requires solving the many-body Schrodinger equation determined by a
Hamiltonian with the form
H =++TTVe h ee + V hh + V eh + V ext (7.4.1)
where Te (Th ) is the sum of the kinetic energies of all electrons (holes),
V,ee V hh and Veh are the sums of electron-electron, hole-hole, and electron-hole
Coulomb two-particle potential energies, and Vext is the external potential, which in our case is the applied THz electric field. Attempting to directly diagonalize this Hamiltonian is a formidable task therefore we will try to simplify the problem in the following way. In the regime of strong quantum confinement the electrons and holes comprising the excitons can be described as uncorrelated particles, therefore in first approximation we can solve the Schrodinger equation including only the kinetic energy terms and consider the terms arising from the Coulomb interaction as a perturbation. The external potential can be made arbitrarily small by varying the strength of the applied electric field, so we can treat this potential as a new perturbation with respect to the many-body states, calculated with the proper account of the Coulomb interaction, and thus estimate the polarizability using a formula similar to Eq. (5.4.1.1).
111 The many-particle exciton states can be constructed in the following manner.
In the absence of any interaction the multiexciton energy levels are given by a sum of terms in the form of Eq. (4.3.1.9), and the wave function is a product of single-particle states given by Eq. (4.3.1.3). The carrier-carrier Coulomb interactions can be written, in second quantized notation, as
1 ee †† VVcccc,ee= ∑∑ abcd a ,s1221 b,s c,s d ,s 2 s,sa,b,c,d12
1 hh †† VVdddd,hh= ∑∑ abcd a,s1221 b,s c,s d ,s (7.4.2) 2 s,sa,b,c,d12 eh †† VVcddceh=−∑∑ abcd a ,s1221 b,s c,s d ,s s,sa,b,c,d12 where s1, s2 are spin variables, c(c)d(d)††, are electron and hole creation
(annihilation) operators, and
Vdd=−rrrrrrrrΨ **()Ψ ()V( )Ψ ()Ψ () (7.4.3) abcd∫∫12 a 1 b 2 12 c 2 d 1 where the single-particle wave functions Ψ represent either an electron and/or a hole, depending on the particular interaction. The Coulomb potential can be written as
ee224π rl V(rr−= ) = < Y* (θ ,φ )Y ( θ ,φ ). (7.4.4) 12∑ l+1 lm 11 lm 22 εεrr12−+l,m 21lr>
These interactions can be treated in the framework of degenerate perturbation theory. Constructing matrices of the different carrier-carrier interactions (Eq.
(7.4.2)) between degenerate multiexciton states, summing and diagonalizing them gives us the energy corrections (the interaction partially removes some of the degeneracies) to the unperturbed many-body state. The eigenvectors give
112 the proper ‘zero’-order wave functions. Due to the complexity of the calculation we are in the process of completing this numerical simulation. The experimental results strongly suggest that the many-body Coulomb interaction plays an essential role in the multiexciton response.
7.5 Conclusions
We have presented a study of the response of multiple excitons in CdSe QDs to an externally applied electric field using the method of optical pump/THz probe spectroscopy. A set of samples with radii ranging from 1.1 to 3.5 nm showed qualitatively similar response when the average number of excitons per QD was varied, consisting initially of a linear response regime followed by saturation. We have shown that the model of uncorrelated non-interacting carriers is adequate only in the limit of small electron-hole pair population and becomes increasingly important as more carriers are introduced. Accounting for the ultrafast process of Auger recombination and including the many-body Coulomb interaction effects is important for the understanding of the multiexciton response in the nanoparticles. This information is important when considering QDs as candidates for various optoelectronic devices.
113 Chapter 8
Size-Dependence of Below-Band-Gap Two-Photon Absorption in
Semiconductor Quantum Dots
The effect of quantum confinement on the nonlinear optical properties of semiconductor quantum dots (QDs) has been a subject of significant interest due to its both fundamental and technological importance [180]. In particular, two-photon absorption (TPA), a third-order nonlinear process, is often used as a complementary method to the linear absorption to investigate the electronic states in QDs because of their distinct selection rules [181]. TPA, when the optical excitation is below the band gap of the QDs, is also essential to QD applications in two-photon microscopy [182] and light up-conversion [183]. Many aspects of the process of TPA have been previously investigated. Experimental studies have explored the spectral dependence of both the degenerate [184, 185] and non-degenerate [186] TPA in a variety of semiconductor QDs embedded in glasses and QD colloids. Both the magnitude and response time [177, 187] of
TPA in QDs have been examined in contrast to their bulk counterparts.
Theoretical calculations of TPA in semiconductor QDs have been less well developed since accurate prediction of the magnitude of TPA requires detailed knowledge of the modified electronic structures of the bulk by quantum
114 confinement. Third -order nonlinearities of QDs have been modeled assuming an interacting electron and hole of effective masses confined in a potential well
[188]. Fedorov et al. [189] first derived an analytical expression for the TPA coefficient in semiconductor QDs based on a parabolic-band effective mass model ignoring the electron-hole Coulomb interaction and band mixing. The model later was extended to the case of non-degenerate TPA and was compared to the spectral dependence of TPA observed in CdTe QDs [186]. The dependence of TPA on the size of the semiconductor QDs, a direct consequence of the effect of quantization in structures of reduced dimensionality, however, is still not well understood. Explicit size dependence can not be easily extracted from realistic theoretical models and both size-dependent [184, 186] and - independent [187] TPA in QDs have been reported.
In this Chapter, we report a systematic study of the size dependence of the degenerate below-band-gap TPA in semiconductor QDs and demonstrate the crossover of the process from atom-like to bulk-like when the QD radius is increased. Transient absorption spectroscopy, as we describe in detail below, was employed to determine the TPA coefficient of CdSe QDs of varying radii.
The TPA coefficient, both its magnitude and variation with QD size was well described by a parabolic-band effective mass model. The model also showed how TPA in QDs approaches that in a bulk in the limit of large dots.
115 8.1 TPA in bulk materials
Investigation of two-photon and multi-photon absorption in various semiconductors acquired a great deal of attention after the invention of the laser, which facilitated the way to obtain monochromatic radiation with very high intensities. The standard experimental technique to measure, for example, the
TPA coefficient of a material, involves recording its transmission at varying fluences. The limited sensitivity of this technique, along with various additional processes such as heating, self-focusing and higher-order nonlinearities, makes extraction of the TPA coefficient difficult and resulted in discrepancies as large as
2 orders of magnitude for the TPA coefficient [77]. Subsequent development of refined techniques, especially that of Z-scan [190, 191], allowed for much more precise measurements and acquisition of reliable data. Theoretical estimation of the effect of TPA usually is based on either a model considering the tunneling effect, or, more commonly, uses the familiar result from second-order perturbation theory. Assuming a simple two-band parabolic model for the electronic structure of the semiconductor allows for the determination of a scaling rule for the TPA
(and multi-photon absorption) coefficient [192]. Assuming that the TPA coefficient is defined through the relation
βω= 2 W/I()22 (8.1.1) where ω is the photon energy, W ()2 is the two-photon transition rate, and I is the intensity, allows us to write this scaling rule as
116 32/ Ep (x21− ) β ===KF(x23 ω /Eg ), F 5 (8.1.2) nEω g (2 x)
where Ep is related to the interband momentum matrix element P (see details
22 below), Ep = 2mP / , nω is the refractive index, Eg is the band gap, and K is a material-independent constant. This expression allows for, at least, an order-of-magnitude estimation, based on the particular material parameters, and shows the correct variation of the TPA coefficient with the frequency [193]. This model, although simple and very useful, does not take into account the specific details of the semiconductor band structure, such as band degeneracy, nonparabolicity, the exciton effect, etc. Inclusion of these effects [194, 195] have shown very good agreement with the experimental data.
8.2 Optical pump/white-light probe technique
An accurate measurement of the TPA coefficient of nanostructures, particularly in composites of low volume fraction, is challenging. The conventional Z-scan method detects the absorption of the excitation beam directly in the open aperture configuration. Its sensitivity is, thus, often limited to ~ 10-3 in the differential transmittance of the excitation beam due to the strong background and other effects such as hot-carrier absorption cannot be separated easily [196].
Here, instead, we will employ the technique of optical pump/white-light transient absorption (TA) spectroscopy. In this method we use TA to determine the TPA
117 coefficient of QDs at the pump wavelength through the pump-induced absorption at the QD first exciton transition, which is calibrated using a pump at a shorter wavelength with a known one-photon absorption cross section. This method is similar to the TPA-induced fluorescence spectroscopy, [181] but by being a time-resolved measurement, as we demonstrate below, allows us to explore the response time and electronic states involved in the TPA process. High sensitivity
-5 (10 in the normalized transient absorption −Δα / α0 ) [197] could be achieved with this method as we describe in detail below.
8.3 Experimental setup and details
The experimental setup consisted of a regenerative amplifier (Spectra Physics) that produces 50-fs pulses centered at 800 nm and at 1 kHz repetition rate. Either the fundamental or the second harmonic of the laser, generated in a 0.5-mm-thick
BBO crystal, was used to excite QD suspensions. A white-light continuum generated in a 2-mm-thick quartz plate was employed as a probe. The pump and probe intersected in a sample of 2-mm thickness non-collinearly at an angle of ~
20°. All measurements were performed with samples at room temperature.
We note that the white light continuum (~ 400-1000 nm) generated through the process of self-phase modulation by focusing femtosecond pulses on a transparent medium such as quartz in this case is highly chirped. It results in significant variation of the arrival time of pulses centered at different wavelengths.
118 For time-resolved measurements it could either be compressed prior to the measurements or be tuned with a monochromator while adjusting the delay time from the pump pulse. The latter has allowed us to perform chirp-free TA spectroscopy with time resolution determined by the duration of the pump pulse.
This approach with high-repetition-rate lasers has achieved the above mentioned
-5 sensitivity of 10 in the normalized transient absorption −Δα / α0 while combined with the lock-in and balance detection technique [197].
On the material system side we investigated CdSe QD colloids as a sample system since size tunable QDs of good uniformity can be routinely synthesized
[91]. We characterized the TPA coefficient of QDs at a fixed excitation wavelength while tuning the QD radius. Series of CdSe QDs, either capped with pyridine or ZnS (NN-Labs) dissolved in 2, 2, 4, 4, 6, 8, 8-heptamethylnonane
(Aldrich) to a typical volume fraction of 10-4 were studied. The QD radius ranged from 1 – 4 nm with a less than ±10% size distribution. The QD size, size distribution, and concentration were inferred from the linear absorption spectra.
Calibration obtained in Ref. [170] agreed with that provided by the QD manufacturer based on TEM measurements, and was used in the interpretation of the linear absorption spectra.
8.4 Results
A typical time dependence of the pump-induced absorption at the first exciton
119 transition of the QDs in the limit of low excitation is illustrated in Fig. (8.4.1). The dashed line represents transient absorption induced by a pump at the fundamental of the laser (800 nm) and the solid line at the second harmonic (400 nm). In this measurement we used ZnS-capped CdSe QDs of 3.3-nm radius.
The linear absorption spectrum is included as an inset. As marked by arrows, the probe was chosen to match the first exciton transition of the QDs at 636 nm and the fundamental and second harmonic of the laser is below and above the band gap, respectively. We note that the first exciton transition of all the QDs investigated here (500 – 640 nm) lies between these two pump wavelengths.
Transient absorption (bleaching) in CdSe QDs induced by 400-nm excitation has been studied extensively [87]. It is known to arise dominantly from state filling.
The fast rise (~ 1 ps) corresponds to ultrafast relaxation from the excited states to the first exciton state of the electron and hole generated through one-photon absorption of the above-band-gap radiation; and the slow decay (~ ns, only the first 40 ps shown in Fig. (8.4.1)) is caused by subsequent carrier trapping and electron-hole radiative recombination. The nearly identical dynamics observed for 800-nm excitation indicates that the below-band-gap radiation injects carriers into QDs practically instantaneously (< 1 ps) which then go through similar relaxation and recombination processes as in the case of 400-nm excitation.
120 -2 6x10 400 nm 800 nm
4
3.5 OPA 3.0 2.5 2.0 Induced absorbance 2 First Absorption Peak 1.5 1.0
Absorbance TPA 0.5 0.0 0 400 500 600 700 800 Wavelength (nm)
0 10 20 30 40 Time delay (ps)
Fig. 8.4.1 Transient absorption dynamics at the first exciton transition of 3.3-nm-radius CdSe
QDs induced by 400-nm (solid line) and 800-nm (dashed line) excitation. Inset: linear
absorption spectrum of the QD suspension; arrows identify the pump and probe wavelengths.
We can, thus, use the absorption cross section of the above-band-gap radiation to calibrate the absorption cross section of the below-band-gap radiation. We note that the effect of different capping on the magnitude of the pump-induced transient absorption was negligible within the first tens of picoseconds after the pump pulse.
It mainly influenced the transient absorption on a much longer time scale due to various trapping sites at the QD surface [198]. In the analysis below we chose the pump-induced transient absorption at ~ 5 ps after the pump pulse to characterize the absorption cross section of the pump to avoid the dynamic Stark effect [87, 133] and the coherent artifacts [199] while carrier trapping and recombination are still not significant. Figure (8.4.2) illustrates a typical
121 dependence of the normalized transient absorption −Δα / α0 at the first exciton transition of the QDs on the pump fluence (symbols).
10 2 3 4 5 6 7 8 9 2 3
3
400 nm 2 Linear fit 800 nm Quadratic fit
0.1
0 9 8
Da/a 7 - 6
5
4
3
2
5 6 7 8 9 2 3 4 5 6 7 8 9 0.1 1 2 Fluence (mJ/cm )
Fig. 8.4.2 Pump fluence dependence of the normalized absorption −Δα / α0 at the first
exciton transition of 2.7-nm-radius CdSe QDs induced by 400-nm (squares) and 800-nm
(triangles) excitation. Solid and dashed lines are linear and quadratic fits.
In the limit of low excitation (with far less than one photogenerated exciton per QD on average), the transient absorption is linearly proportional to the fluence of the
400-nm excitation F2ω (solid line) and quadratically proportional to the fluence of
2 the 800-nm excitation Fω (dashed line): −==Δα / α0122CFωω CF , where C1 and
C2 are the fitting parameters. The fluence dependence, together with the absorption dynamics, is a direct evidence of simultaneous TPA of the below-band-gap excitation in the QDs. The magnitude of the transient
122 absorption, on the other hand, is determined by the average exciton number in a
−α()1 L dot, and in the low excitation limit, linearly as −=Δα / α02CF()ω /τ (e1 − ) =
2 ()2 (C/2 )() Fω / ταL . Here parameter C is determined by the experimental details; the factor 2 takes into account that twice as many absorbed photons are required to generate the same amount of excitons through two-photon than one-photon absorption; α(i), i=1,2 are the one- and two-photon absorption coefficient of the QD suspension of thickness L , which are defined as usual as the linear and quadratic attenuation coefficient of the peak excitation intensity with distance of propagation. In writing this relation, we have ignored depletion of the below-band-gap pump and assumed the pulse duration of both the fundamental and second harmonic of the laser to be τ . The TPA coefficient of the QD suspension can, thus, be extracted from Fig. (8.4.2) as
()1 2C τ (e1− −α L ) α()2 = 2 (8.4.1) CL1
We now convert the TPA coefficient α(2) of the QD suspension to the TPA cross section σ ()2 of an individual QD, an intrinsic property of the QD, through
(2)4 (2) α = 2/Nfω σω . Here N , , and ω denote the QD concentration in the suspension, Planck’s constant, and the angular frequency of the excitation,
respectively; fω =+()/(2)ε + 2ε h εεihis the local-field factor, determined by the
dielectric constants of the suspension ε , the QD εi and the solvent εh . The extracted TPA cross section of CdSe QDs of radii 1 – 4 nm is shown in Fig.
123 (8.4.3(a)) (circles). The error bars on σ()2 reflect the reproducibility of the measurements and the effect of QD size distribution. There is an overall uncertainty of a factor ~ 2 in the vertical scale associated with the experimental determination of the pump pulse duration and the beam size.
60 60 -45 1.0x10 (a) (b) Model 50 50 0.8 40 40 (cm/GW) s) β 4 0.6 30 30 (cm
(2) 0.4 σ 20 20
0.2 10 10 TPA coefficient 0.0 0 0
1.0 2.0 3.0 4.0 0 1 2 3 4 Radius (nm) Radius (nm) Bulk
Fig. 8.4.3 (a) Dependence of the two-photon absorption cross section σ()2 at 800 nm on the
QD radius; (b) Dependence of the two-photon absorption coefficient β at 800 nm on the QD
radius. Circles: experiment; solid line: parabolic-band effective mass model as described in
the text; triangles: β of bulk CdSe at 800 nm.
The derived TPA cross section σ()2 of individual QDs has a magnitude of ~ 10-46 cm4s, which far exceeds 10-49 cm4s, a typical value for an atom or molecule with a single TPA resonance [200]. This is a direct consequence of the large size of the
QDs compared to that of an atom. And by the same token, σ ()2 is seen to rise very sharply with the increase of the QD radius. Intuitively, TPA at a fixed
124 excitation wavelength is expected to become bulk-like and σ()2 to scale with the
QD volume V in the limit of large QDs. Therefore, we define a TPA coefficient
β as β = 2/()σω(2) V and compare it to the bulk value in CdSe [201] (triangles) in Fig. (8.4.3(b)). The TPA coefficient of QDs shows the same order of magnitude as that of a bulk. It increases with the size for QDs of radius ranging from 1 - 4 nm with possible resonance at certain radii, a characteristic often observed for atoms or molecules. No significant saturation is observed.
8.5 Model for TPA in QDs
To understand how the TPA coefficient in QDs depend on the dot radius and how it approaches that of a bulk, we consider an effective mass model starting with evaluation of a two-photon transition rate W ()2 in a QD based on the second-order perturbation theory:
2π 2 WM()2 =−−δ(E E 2 ω ), (8.5.1) ∑ b,b10 b 1 b 0 b,b10
HHb,b b ,b M = 12 20 (8.5.2) b,b10 ∑ EE−−− ω i γ b2 bb20 b 2
Here b0 , b1 , and b2 represent the initial, final, and intermediate state of the
system, Eb and γb denote the energy and broadening of the b -th state, and the summation is extended to all initial and final states that satisfy the energy conservation relation. We take the interaction H to be of the form (e/mc)A⋅ p , where e and m are the elementary charge and mass of an electron, c is the
125 speed of light in vacuum, A denotes the vector potential associated with the pump radiation, and p is the carrier momentum operator. The electronic wave function of a QD can be expressed as a product of a cell-periodic Bloch function, similar to that in a bulk, and an envelop function. Such a modification leads to a TPA process in QDs that consists of an interband and an intraband transition and bears both atomic and bulk characteristics. Since a realistic electronic structure for CdSe QDs resulting from diagonalizing an 8−band
Hamiltonian [130] is complicated to apply for the evaluation of the two-photon transition rate of Eq. (8.5.1), we adopted the parabolic-band effective mass model
[189] for simplicity. In this model all 8 bands are considered, but band mixing and exciton effects are ignored and an infinite spherical square well is assumed at the boundary of the QD. The envelope part of the wave function can be expressed with the help of Eq. (4.3.1.3), while the cell-periodic Bloch part is given for the ‘heavy’ and ‘light’ hole bands, by
1 1212/,/=−↓+↑ (XiY) Z , 3 (8.5.3) i 12/,− 12 /=−−↑+↓ (XiY) Z 3 for the J/= 12 spin-orbit split-off band, and by
12/,± 12 /=↑↓ iS (8.5.4) for the conduction band, where S is a totally symmetric function. The energy levels for the conduction and the valence bands are given by
126 22ζζ 22 E,c ==−−nl EEhj nl (8.5.5) nl22mR22 nl hj m R chj where E = E= 184. eV, E= 226. eV, and the effective masses are hh12 h 3 m = 011. m, m = 119. m, m = 0. 403 m, and m = 0602. m. Evaluation of c h1 h2 h3 the interband transition resembles that of bulk, resulting in a transition determined by the overlap of the envelope functions, multiplied by the momentum matrix element between the corresponding Bloch states. For the 3 possible valence-to-conduction band transitions this leads to
PeA ⎛⎞3e 0 2e e−1 −e0 − 2e c(e/mc)hAp⋅= iδδδ +1 0 −1 n'l' m' nlm n' n l'l m' m ⎜⎟ 32 cee⎝⎠0 32ee−+1100 e + 1
(8.5.6) where P is the interband matrix element of the electron momentum
22 ( 221mP / ≈ eV [202] for most semiconductors), and e(eie)/±1 =±∓ xy 2 ,
ee0 = z are the spherical components of the unit polarization vector. The intraband transitions can be handled with the help of Eq. (5.4.1.2) and (5.4.1.3).
Combining this with Eq. (8.5.6) the orientation-averaged TPA coefficient for QDs of cubic lattice symmetry, normalized by the volume fraction, is found to be
128π324Pe 3 β = F (8.5.7) 22 3∑ c,hj 3Vcnωω j=1
22 1 c,hζζ h FT(l=+j δ l δ ) nl11 nl 0 0 δ(Ec −− E j 2 ω ) c,hj R(2222∑ nl,nl11 0 01101 l,l 1 0+− l,l 1 0ζζ− ) nl 11 nl 0 0 nl11 ,nl 0 0 nl11 nl 0 0
(8.5.8) where
127 2
c,hj 11 11 T.nl ,nl =+ 11 0 0 m(Ecc−+− E ω i γ c)m(Ehhjj−++ E ω i γ h j) cnlnl00 11 nlh 00 j nl0 0 nl 11 nl 11
(8.5.9)
QD size distribution can be further taken into account by averaging F over a c,hj normalized distribution function f (R) to yield
ζζ22 f (R(hj ) ) 1 c,hj nl11 nl 0 0 nl 11 ,nl 0 0 FT(lc,hj =+δ l δ ) ∑ nl11 ,nl 001101 ll 10+− ll 10 222(hj ) 22( ω −−E)nl ,nl (ζζ) R hnlnlj 11 0 0 11 0 0 nl11 ,nl 0 0
(8.5.10)
2 ⎛⎞2 2 (h ) ζ ζ with R j =+⎜⎟nl11 nl00 . To evaluate Eq. (8.5.10) numerically nl11 ,nl 0 0 22( ω − E)m⎜⎟ m hchjj⎝⎠ we assumed a Gaussian size distribution with a 10% standard deviation. The level broadening was assumed to be 60 meV [203] for all states. To ensure convergence of the sum in Eq. (8.5.10) we included for all QDs excited conduction and valence states 12 and 5 eV, respectively, from their ground states.
The model (solid lines) is compared with experiment in Fig. (8.4.3). The agreement is surprisingly good considering the measurement uncertainties, the simplification in the model and non-cubic lattice symmetry of CdSe QDs. The model underestimates the TPA coefficient β in small QDs due to the neglect of the exciton effects and TPA resonances have been largely depressed due to the
QD size distribution.
In the limit of large QDs, the summation in Eq. (8.5.10) can be replaced by
128 integration. The actual distribution of the energy levels is complicated, but can be simplified when nl using the asymptotic formula for the zeros of the spherical Bessel functions [204]. It is well-known that in this limit the density of
states (to which F c,hj is proportional) depends only on the volume, which results in a size-independent TPA coefficient, as expected.
8.6 Conclusions
In summary, TPA in CdSe QDs of varying radii at 800 nm (below band gap) is found to be instantaneous. The TPA coefficient β of small CdSe QDs ( R =14− nm) with possible resonance at certain radii increases with the dot size due to the increase of the number of allowed final states. The theoretical model indicates that β saturates and becomes nearly size independent when R > 5nm. And in the limit of large dots discrete levels merge into quasi-continuum and β approaches its bulk value.
129 Chapter 9
Summary
In summary, we have presented some applications of terahertz time-domain spectroscopy and used this technique, in conjecture with an ultrafast optical excitation, to study the response of photogenerated carriers in semiconductor quantum dots.
We investigated the generation of THz radiation through optical rectification in a highly-focused geometry, demonstrating favorable power throughput at sub-wavelength resolution. This study, which has possible applications in near-field imaging, was numerically simulated by a model that accounts for the specifics of the generation process, as well as for the propagation of the long-wavelength radiation through air and various optical components.
An extension of this model was used to explain the effect of finite beam sizes in an optical pump/THz probe experiment, a common issue in typical experimental geometries. We have demonstrated that non-trivial spectral deformations occur when the geometry of the experiment is not optimal. Based on the numerical calculations we have proposed a criterion that can be used as a guideline in real measurements, with the possibility to correct for less-than-perfect experimental conditions.
130 One of the truly unique features of THz-TDS is the ability to perform spectroscopic measurements in the FIR with a picosecond resolution. This allowed us to study the response of short-lived photoexcited carriers to an external electric field in semiconductor quantum dots. A comparison study of the single-exciton polarizability in two different systems of QDs revealed an ‘atom’-like response, characteristic of well-separated energy levels, which scaled as the fourth power of the radius. The results were simulated with the help of standard electronic structure models, based on the effective-mass approximation.
The time resolution of the THz spectrometer was further exploited in a study of the ultrafast Auger recombination in CdSe QDs, a relaxation channel that is dominant in the case of multiple carriers confined in a small volume. Our results confirmed previous works, regarding the typical recombination lifetimes, indicating that using THz-TDS, a non-resonant technique for this system, is an adequate tool for monitoring the relaxation dynamics in QDs.
The response of multiple excitons to an applied THz electric field in CdSe
QDs was investigated, showing qualitatively similar behavior in samples of different size: a linear increase of the polarizability was followed by saturation when more and more carriers were photoexcited. An account of the Auger recombination and the Coulomb many-body effects explains the observed trend.
Finally, we studied the dependence of the two-photon absorption coefficient
(at a fixed wavelength) in semiconductor quantum dots with different size, with the
131 help of the optical pump/white-light probe technique. Our results indicate a response that increases with the sample size, a phenomenon that can explained with the increase of the density of allowed final states. A numerical simulation based on the simple parabolic-band model reproduces the observed behavior.
In this thesis we have shown that the method of THz-TDS provides useful information when applied to study semiconductor quantum dots. The discreteness of the electronic energy levels, much larger than the THz photon energy, indicates that this is a non-resonant method that measures the real response of photoexcited carriers to THz electric field. A large area of possible applications of THz-TDS lies in the domain of studying the resonance phonon properties of various systems of QDs. One could imagine that an extension of the useful frequency bandwidth can easily provide valuable spectroscopic information, including size- and temperature variation of quantized phonon modes, dynamics due to various excitations, strength of exciton-phonon coupling, etc.
This area can gain further interest with the development of more intense sources of THz radiation that can lead to pump/probe experiments performed exclusively in the far-infrared.
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