Department of Physics of the College of Arts and Sciences

November 15, 2006

Suvranta K. Tripathy

Doctor of Philosophy (PhD)

Dephasing of excitons and phase coherent photorefractivity in ZnSe quantum wells

Professor Hans-Peter Wagner Professor Paul Esposito Professor Leigh Smith Professor Rohana Wijewardhana

1 Dephasing of excitons and phase coherent photorefractivity in ZnSe

quantum wells

A dissertation submitted to the

Division of Research and Advanced Studies

of the University of Cincinnati

In partial fulfillment of the

requirements for the degree of

DOCTORATE OF PHILOSOPHY (PhD)

In the Department of Physics

of the College of Arts and Sciences

2006

Suvranta K. Tripathy

M.S. (Physics), Utkal University, India, 1999

Committee chair: Professor Hans-Peter Wagner

1 Abstract

Spectrally resolved and time-integrated degenerate four-wave mixing (FWM)

experiments with ultra-short pulses have been used to investigate the relaxation times and

coherent interactions of excitons in the Boltzmann and in the quantum kinetic regime in

ZnSe single quantum wells. Three-pulse FWM experiments using 100 fs pulses in a 10

nm thick ZnSe/ZnMgSe single quantum well reveal nearly equal zero-density dephasing

rates for coherences between σ + and σ − heavy- and/or light-hole exciton states as compared to the dephasing rate of the interband coherence between these states and the ground state. The measurements also demonstrate that exciton–exciton scattering destroys the phase coherence between heavy- and light-hole excitons about three times more effectively than the coherence between σ + and σ + heavy-hole excitons. The

different coherent nonlinear signals obtained in a three-beam FWM experiment are

explained and reproduced by numerical calculations of the optical Bloch equations for a

ten-level model including excitation induced dephasing (EID).

Two-beam FWM experiments using 30 fs pulses on a 10 nm ZnSe quantum well

reveal marked quantum beats and a pronounced non-exponential decay for delay times

smaller ~500 fs. The pronounced peak at pulse overlap is attributed to rapid destructive

interference of the created polarization and to spin dependent quantum kinetic effects

caused by the transient formation of an electron-hole ( eh ) pair grating. The three-beam

FWM trace shows a bi-exponential decay which is attributed to a nonlinear polarization

that is caused by the interaction of polarization k3 with both an exciton grating and an

eh -pair grating. Model calculations using the OBEs of a two level system that includes

2 EID at both types of density grating provide information about exciton formation processes involved. Model calculations assuming geminate exciton formation reveal an

γ −1 = exciton formation time of R 5 ps . Intensity dependent experiments show exciton-

β = ± -1 2 eh -pair scattering parameter of X −eh 90 20 s cm .

We further observe coherent exciton-LO-phonon polarons in the FWM spectrum

of a 3 nm ZnSe single quantum well using 30 fs light pulses. The formation of the new

quasi-particles is attributed to a strong phonon coupling that is caused by strong Fröhlich

interaction, a huge exciton binding that exceeds the LO-phonon energy and a close

resonance between the 2s exciton state and LO-phonon energy. The rapid decay of these

coherent quasiparticles is attributed to the disintegration into zero-phonon excitons and

free LO-phonons as well as to the inhomogeneous broadening of LO-phonon energies

due to disorder and k-space dispersion

A novel phase coherent photorefractive (PCP) effect has been observed in ZnSe single QWs using ultra-short pulses that do not overlap in time and its spectral and thermal dependence have been investigated. The observed PCP effect is attributed to the formation of an electron grating formed by the interference of coherent excitons. The experimental results are reproduced by a phenomenological model that is based on optical Bloch equations for a five-level system. The high efficiency of this PCP detectable at an average power of a few µW makes it attractive for applications in all-

optical data processing, optical coherent imaging and optical switch applications.

To my parents, wife and daughter

5 Acknowledgements

I take this opportunity to extend my gratitude to the people whose contribution

helped to achieve my doctoral dissertation. I would like to thank all faculty, staff, and

students at the department of physics, University of Cincinnati for their support

throughout my PhD life.

My special thanks go to my thesis advisor Prof. Hans-Peter Wagner, who is solely responsible for completing this challenging research work. His new experimental and theoretical ideas and stimulating discussions, his continuous support and encouragement facilitated me to complete my PhD. It has been a pleasure to work with him and I will always be grateful to him for his guidance that brought me up to this position.

I would like to thank Prof. Paul Esposito, Prof. Leigh Smith and Prof. Rohana

Wijewardhana for serving on my doctoral committee. I am obliged to Prof. Rohana

Wijewardhana who was always available to help me to solve problems and to prepare me for the qualifying examination and also for his support in my graduate career in various ways. I am thankful to Prof. Leigh Smith who always gave the confidence and advice to work hard. We are thankful to Prof. D. Hommel for providing us 3 nm ZnSe QW samples.

A special acknowledgement is extended to Dr. Hans-Peter Tranitz for his training in ultra-fast lasers, in the FWM experiment and for his explanations, as well as for being a good friend. I want to extend my gratitude to John Markus, the instrumentation specialist at the Department of Physics for his help at many occasions in the lab work. A big thank to Melody Whitlock (former secretary), Donna Deutenberg, Elle Mengon, Bob

Schrott, Mark Ankenbauer, Mark Sabatelli, and John Whitaker. Many thanks to my lab-

6 mates Ajith Desilva, Pradeep Bajracharya, Venkat Gangilenka and Amin Kabir for sharing their talents, ideas and humors and for being a part of my PhD life.

I would like to thank all of my teachers from Utkal University especially to Prof.

Puspa Khare, Prof. Niranjan Barik, Prof. Lambodar Prasad Singh, Prof. Karmadev

Maharana, and Prof. Swapna Mahapatra for their endless encouragement and support to have a carrier in physics.

My parents have been my strongest source of inspiration. Their blessing, love and emotional support led me to this success in my life. I am indebted to my brothers and sisters whose constant support helped me to come to this point. I express my admiration and deepest gratitude to my wife Meena for her continuous support and understanding, for her love and sacrifice that led me to achieve my dreams. Also I would like to thank my beautiful daughter Selina for being a part of my life who brought lot of happiness during my PhD life.

7 Thesis outline

1. Introduction …………………………………………………………………………16

2. ZnSe based semiconductor quantum wells ………………………………………..20

2.1. Electronic band structure……………………………………………………...... 20

2.2. Excitons…………………………………………………………………………22

2.3. Linear and non-linear optical properties……………………………………...... 29

2.4. Electro-optical properties and photorefractive effect………………………...…31

3. Dephasing of excitons : Theoretical background …………………………………34

3.1. Dephasing of excitons: Boltzmann regime …………………………………...34

3.1.1. Principle of four-wave mixing (FWM)………………………………...... 35

3.1.2. OBE for a homogeneously broadened system………………………...…38

3.1.3. OBE for an inhomogenously broadened system…………………………39

3.1.4. Various FWM processes…………………………………………………41

3.1.5. Exciton-exciton scattering……………………………………………….45

3.1.6. Exciton-LO phonon interaction………………………………………….46

3.2. Extended OBEs: Ten level model …………………………………………….47

3.3. Brief introduction into quantum kinetics ……………………………………50

4. Description of the experimental setup ……………………………………………..53

4.1. Sub-30 fs laser: Theory and operation ………………………………………..53

4.1.1. Laser pulses………………………………………………………………53

4.1.2. Group velocity dispersion (GVD) and its compensation………………...54

4.1.3. Principle of mode-locking………………………………………………..56

4.1.4. Sub-20 fs Ti: Sapphire laser design……………………………………...57

4.2. Pulse width measurement: Autocorrelation technique ……………………..61

8 4.3. Four-wave mixing experimental setup ………………………………………..67

5. Four-wave mixing experimental results ...... 69

5.1. Study of interband and interexciton coherences ……………………………..69

5.1.1. Three-beam FWM experiments………………………………………….69

5.1.2. Theoretical model………………………………………………………..73

5.1.3. Intensity dependent measurements………………………………………78

5.2. Carrier correlated dephasing of excitons in 10 nm ZnSe QW...……………84

5.2.1. Two-beam experiments…………………………………………………..84

5.2.2. Three-beam experiments………………………………………………....92

5.3. Observation of exciton-LO phonon polarons in a 3 nm ZnSe QW………....98

5.4. Phase coherent photorefractive (PCP) effect ……………………………….108

5.4.1. Two-beam FWM measurements; Observation of PCP……………...…108

5.4.2. Three-beam FWM measurements of PCP effect……………………….115

5.4.3. Spectral dependence of the PCP effect…………………………………120

5.4.4. Temperature dependence of the PCP effect…………………………….126

5.4.5. PCP with cw pump laser………………………………………………..128

6. Summary …………………………………………………………………………...131

7. References ………………………………………………………………………….135

9 List of figures

Figure 1: Schematic band structure of ZnSe at the center of the Brillouin Zone showing the angular momentum of valence and conduction band and the heavy-hole band and light-hole band away from the center. 21

Figure 2: Schematic diagram of energy band structure of type-I quantum wells. 25

Figure 3: Schematic diagram of a finite potential well of depth V0 given by equation (10) after [1]. 26

Figure 4: Schematic absorption spectra of a two dimensional semiconductor which shows each step of the step function acquires its own excitonic series, after [1]. 27

Figure 5: The diagram depicts the allowed optical transitions for the HH and LH excitons, after [9]. 28

Figure 6: Parallel and perpendicular geometries of a photorefractive quantum well structure, after [15]. 32

Figure 7: Schematic diagram of a three-beam FWM experiment. 36

Figure 8: Diagram showing the phase mismatch and the phase matching direction in a 2- beam FWM experiment. 37

Figure 9: The model shows the formation of the biexciton through two opposite circularly polarized light pulses. 44

Figure 10: Ten-level model. 47

Figure 11: TI-FWM from GaAs at 77 K using 14.2 fs pulses for three different carrier densities that shows the beating between interband polarizations connected by LO phonon scatterings, from [58]. 51

Figure 12: Schematic diagram for GVD compensation using two prisms, after [69]. 56

Figure 13: Schematic diagram of the sub-20 fs laser. 59

Figure 14: Schematic diagram of an interferometric autocorrelator. 62

Figure 15: Experimental setup of the two-photon absorption autocorrelator used to measure the pulse width of sub-50 fs pulses (left figure). Schematic diagram of applied saw tooth voltage to the PZT (right figure). 64

10 Figure 16: Applied saw-tooth voltage that shows the actual applied voltage and the measurement of TSweep . Autocorrelation trace of the frequency doubled 23 fs fundamental pulse. Using an autocorrelation factor of 0.65 for sech 2 pulses the frequency doubled pulses have a temporal pulse width of 38 fs. 66

Figure 17: Schematic diagram of the experimental setup for FWM experiment. 67

Figure 18: Spectrally resolved 3-beam FWM signals for polarization configurations(σ +σ +σ + ) , (σ +σ +σ − ) and (σ +σ −σ + ) at 55K, taken at a delay time τ = - 0.55 ps. 69

Figure 19: FWM traces at the spectral position of the heavy-hole Xh exciton transition for different polarization configurations and at the heavy-hole biexciton induced transition + − + XX h for configuration (σ σ σ ) at 55 K. 71

+ − Figure 20: FWM traces into k 3 k 2 k1 direction at the spectral position of the light- + + + + + − hole Xl exciton transition for polarization configurations (σ σ σ ) , (σ σ σ ) and (σ +σ −σ + ) at 55 K. 73

Figure 21: Calculated 3-beam FWM spectra for polarization configurations(σ +σ +σ + ) , (σ +σ +σ − ) and (σ +σ −σ + ) for the same conditions as in fig. 18. 74

Figure 22: Calculated 3-beam FWM traces at the energetic position of the heavy-hole Xh exciton transition for configurations (σ +σ +σ + ) , (σ +σ +σ − ) and (σ +σ −σ + ) and at the + − + biexciton induced transition XX h for configuration (σ σ σ ) for the same conditions as in fig. 19. 75

Figure 23: Calculated 3-beam FWM traces at the energetic position of the light-hole Xl exciton transition for configurations (σ +σ +σ + ) , (σ +σ +σ − ) and (σ +σ −σ + ) for the same conditions as in fig. 20. 77

+ − + Figure 24: (a) Normalized 3-beam FWM traces for (σ σ σ ) configuration at the Xh exciton transition for a fixed k1 and k 2 pulse intensity and variable k3 pulse intensity labeled as ratio I 3/I 03 . (b) Measured dephasing rates from FWM traces at the Xh, XX h and

Xl transition energy as a function of the exciton density N3 generated by pulse k3 The γ dashed line gives a linear increase of the dephasing rate 21 with an exciton-exciton β = −1 2 scattering parameter of 21 22 s cm . 79

+ − + Figure 25: (a) Normalized 3-beam FWM traces for (σ σ σ ) configuration at the Xh exciton transition for a fixed k 3 pulse intensity and variable k1 and k 2 pulse intensity

11 labeled as ratio (I 1+I2)/ (I 01 +I 02 ). (b) Measured dephasing rates from FWM traces at the Xh, XX h and Xl transition energy as a function of the exciton density N1+N 2 generated by pulses k1 and k 2 . The dashed and dotted-dashed lines give a linear increase of the γ γ β = −1 2 dephasing rate 32 and 72 with scattering parameter 32 22 s cm β = −1 2 and 72 60 s cm , respectively. 81

Figure 26: Two-beam FWM spectrum (full line) of a 10 nm thick ZnSe SQW at delay τ ≈ ↑↑ 12 0 at 55 K using 30 fs excitation pulses in ( ) configuration. Excited exciton transitions are labeled. The dashed line shows the reflection spectrum of the incident laser pulse. The peak energy was centered at 2.805 eV. The intensities of pulses k1 and k 2 were 2.2 and 3.0 MW/cm 2, respectively. 84

Figure 27: FWM at the spectral position of the 11h exciton transition for polarization configuration ( ↑↑) at 55 K. Also shown is a calculated FWM trace based on a multi-level model (thin full line). The inset shows the Fourier transformed spectrum of the FWM trace. Different exciton transitions are labeled. 85

Figure 28: FWM traces at the spectral position of the 11h exciton transition for collinear ↑↑ ↑→ ( ) and cross-linear ( ) polarized fields at 55 K. The intensity of pulses k1 and k 2 were 2.2 and 3.0 MW/cm 2, respectively. 88

Figure 29: Normalized FWM traces at the spectral position of the 11h exciton transition ↑↑ for polarization configuration ( ) at 55 K with k1 pulse intensities ranging from 0.5 to 8.1 MWcm -2. The inset shows extracted dephasing rates as a function of the eh-pair

density created by pulse k1 . The dashed line gives a linear increase of the dephasing rate γ 2 according to eq. (53) using an exciton-eh-pair scattering parameter of β = −1 2 X −eh 90 s cm . 90

+ − Figure 30: FWM trace into k 3 k 2 k1 direction at the 11h exciton transition as a τ σ +σ +σ + τ = function of delay 13 for polarization configuration ( ) at 12 0 (open circles). 2 The intensities of pulses k1 , k 2 and k 3 were 1.3, 1.3 and 1.0 MW/cm , respectively. The center of the pulse energy was set to 2.805 eV. Also shown are results of model calculations (full, dashed, dotted-dashed lines) as described in the text. The inset shows τ τ = the FWM trace as a function of delay 12 but for fixed 13 10 ps. 93

Figure 31: Photoluminescence spectrum (full line) and photoluminescence excitation spectrum (dotted dashed line) of a 3nm ZnSe quantum well sample as described in the text. Also shown are reflectivity spectra using a Hg lamp (open circles) and a 35 fs light pulse (dashed line) used in the FWM experiments. All spectra were taken at 30 K. 98

12 Figure 32: Contour plot (logarithmic scale) of the FWM signal of a 3 nm ZnSe QW ↑↑ τ obtained in collinear ( ) configuration as a function of delay 12 between pulses k1 and 2 k 2 with center energy 2.9 eV. The total pulse intensity is 15 MW/cm . Also shown is the FWM spectrum at τ12 = 100 fs on a logarithmic scale. 99

+ + Figure 33: Normalized FWM spectra obtained in co-circular ( σ σ ), collinear ( ↑↑ ) and cross-linear ( ↑→ ) configuration, as labeled, at pulse delay τ12 = 0 and pulse center energy 2.9 eV. The total pulse intensity is 15 MW/cm 2. The spectra are offset for better comparison. 102

Figure 34: Normalized FWM spectra obtained in collinear ( ↑↑ ) configuration at different delay τ12 , as labeled. The pulse center energy is 2.915 eV; the total pulse intensity is 35 MW/cm 2. The spectra are offset for better comparison. 103

Figure 35: FWM traces at the spectral position of the heavy-hole Xh exciton and at mixed polaron energies X h+1LO and X h+2LO on a logarithmic scale. The pulse center energy is 2.915 eV; the total pulse intensity is 35 MW/cm 2. The inset shows the Fourier transformed spectra of the FWM traces at positions X h and X h+1LO, respectively. 104

Figure 36: Contour plot (logarithmic scale) of the FWM signal obtained in cross-linear ↑→ τ ( ) configuration as a function of delay 12 between pulses k1 and k 2 with center energy 2.9 eV. The total pulse intensity is 15 MW/cm 2. Also shown is the FWM spectrum at τ12 = 15 fs on a logarithmic scale. 105

Figure 37: FWM traces at the spectral position of the X h exciton and at the light-hole related “X l” transition on a logarithmic scale. The pulse center energy is 2.9 eV; the total pulse intensity is 15 MW/cm 2. The inset shows the Fourier transformed spectrum of trace at position X h. 106

Figure 38: Dependence of the diffraction efficiency at the spectral position of the 11H Xh exciton transition recorded at 55 K. The dashed line shows a fitted curve as described in 2 the text; the solid line shows an I p dependence. The inset demonstrates intensity dependent FWM traces at different k1 and k 2 pulse intensities as labeled. ( I0 = 800 kWcm -2) 108

Figure 39: Schematic diagram (left) shows the steps for formation of the electron grating in the quantum well after the decay of the exciton grating. The right figure shows the transfer of electron from the GaAs substrate to the quantum well. 110

− τ Figure 40: Spectrally resolved PR signal into 2k 2 k1 direction as a function of delay 12 recorded at a lattice temperature of 55 K. The total intensity of the co-circular polarized -2 pulses is 20 kWcm . The inset shows the calculated trace at the Xh heavy-hole exciton

13 transition as described in the text (full line) in comparison to the experimentally observed trace (dashed line). 112

+ − Figure 41: Traces of the diffracted k 3 k 2 k1 PR signal at the Xh exciton transition for polarization configurations (σ +σ +σ + ) , (σ +σ −σ + ) and (σ −σ −σ + ) recorded at 55 K. The inset shows the spectrally resolved contour plot for polarization configuration (σ +σ +σ + ) . τ = In addition the PR signal spectrum at delay 13 100 fs (solid line) and the spectrum of the 90 fs excitation pulse (dashed line) is displayed. 116

+ − Figure 42: Calculated traces of the diffracted k 3 k 2 k1 PR signal at the Xh exciton transition for polarization configurations (σ +σ +σ + ) , (σ +σ −σ + ) and (σ −σ −σ + ) for the same conditions as in Fig. 47. The inset shows the calculated spectrally resolved contour plot for polarization configuration (σ +σ +σ + ) and the PR signal spectrum at delay τ = 13 100 fs. 119

− Figure 43: Traces of the diffracted 2k 2 k1 PR signal at the Xh heavy-hole exciton τ transition as a function of delay 12 recorded at a lattice temperature of 55 K and at different excitation energies as labeled. The pulse intensities of co-circular polarized = = −2 pulses k1 and k 2 are I1 80 , and I2 40 kWcm , respectively. 120

− τ > Figure 44: Spectra of the diffracted 2k 2 k1 PR signal for positive delay 12 0 recorded at a lattice temperature of 55 K and at different excitation energies as labeled. The dashed lines show the spectra of the excitation pulses for comparison. The used pulse intensities are the same as in Fig. 43. 121

τ Figure 45: PR signal intensities (a) and energy shift at 12 = 500 fs as well as exciton dephasing rates γ2 of the Xh heavy-hole exciton transition (b) as a function of the excitation energy. 122

− τ > Figure 46: Spectra of the diffracted 2k 2 k1 PR signal for positive 12 0 (solid line) τ < and negative delay 12 0 (dashed line) recorded at a lattice temperature of 55 K and at different excitation energies as labeled. The inset shows the obtained redshift of the Xh heavy-hole exciton transition as a function of the captured electron density ne evaluated from eq. (66). The used pulse intensities are the same as in Fig. 43. 125

− Figure 47: Traces of the diffracted 2k 2 k1 PR signal at the Xh heavy-hole exciton τ transition as a function of delay 12 recorded at excitation energy 2.808 eV and at different lattice temperatures as labeled. The co-circular polarised pulse intensities are = = −2 I1 40 , and I2 30 kWcm , respectively. The inset shows the PR signal intensity τ = obtained at 12 500 fs as a function of temperature. 126

+ − Figure 48: Traces of the diffracted k 3 k 2 k1 PR signal at the Xh heavy-hole exciton transition for polarization configurations (σ +σ +σ + ) , (σ −σ −σ + ) and (σ +σ −σ + ) as a τ function of delay 12 recorded at a lattice temperature of 25 K. The delay time between = k1 and k 3 was kept fixed to 10 ps. The intensities of pulses k1 , k 2 and k 3 were I1 34 , = = −2 I 2 24 and I3 60 kWcm , respectively. 127

− Figure 49: Traces of the diffracted 2k 2 k1 PR signal at the Xh exciton transition for polarization configurations ( σ+σ+) recorded at 55 K for different spatially uniform cw laser illumination of the GaAs substrate at 637 nm (1.93 eV). The increasing intensity ratio is given with respect to a reference power of P 0 = 50 µW. The inset shows the exciton dephasing rate γ2 (open squares) as well as the observed PRE signal intensity at τ 12 = 500 fs (full circles) as a function of the cw laser power. 129

15 1. Introduction

The development of fast optoelectronic devices and future technology is based on

the understanding of the optical properties in semiconductors and semiconductor

nanostructures. Semiconductor lasers, light emitting diodes, all-solid light detectors and

electro-optical as well as all-optical switching devices are examples that illustrate the

importance of the semiconductor optics. The application potential of semiconductors for

optical and electronic purposes is further enhanced by miniaturizing the device size to a

few nano-meters and by producing new material combinations of extremely small

dimensions with predictable characteristics. In semiconductor nanostructures such as

quantum wells, the quantum mechanical wave functions of electrons and holes are

confined inside the material, giving rise to the so called quantum-confinement effects that significantly modifies optical properties [1]. In such nano-structures electrons have to be considered as waves and the coherence between them can be used to develop novel electronic and photonic devices that are essential for future computation and telecommunication.

A fundamental step forward to this achievement is to investigate decoherence time of electronic transitions in semiconductor nanostructures and to attain a profound understanding of the dephasing processes involved. This knowledge is essential to modify these processes e.g. by quantum confinement effects and finally find solid state systems with long-living coherent states possibly suitable for quantum computing. A powerful tool to study such quantum coherent processes is the ultra-fast spectroscopic method of degenerate four-wave mixing (FWM). The FWM experiments can be performed in two-beam and three-beam configurations. The FWM experimental

16 technique performed at different excitation conditions e.g. at different intensities of incident laser beams or at varying sample temperature enable the study of exciton-exciton interaction, interaction of exciton with free carries and exciton-phonon interaction respectively. Polarization dependent measurements enable to control the FWM signals arising from different excitonic transitions. The three-beam configuration enables to determine the lifetime of different excitonic transitions.

In the past these investigations were mainly performed on III-V semiconductor nanostructures. II-VI semiconductors qualify as an ideal material to investigate fundamental quantum coherent processes because of the higher exciton oscillator strength and huge exciton binding energy. In addition II-VI semiconductor possesses high biexciton binding energy that enables a spectral separation of biexciton induced signals.

This thesis presents the study of dephasing times of different excitonic transitions and their correlations in high-quality ZnSe single quantum wells (QWs) of different well widths. These investigations have been performed using the FWM experiments on a 100 fs time scale being appropriate to analyze the dephasing of coherent excitons due to

Coulomb-interaction or due to phonon scattering in the dissipative regime. These processes are explained by 10-level extended optical Bloch equations in the limit of semi- classical Boltzmann kinetics.

In addition a new fascinating and important research field has been enforced using a home built ultra-short sub-30 fs laser. These measurements concern the coherent coupling between excitons and LO-phonons where the dynamics can be explained by non

Markovian quantum kinetics . In this context the correlation of electrons and holes on the

dephasing of excitons on a ZnSe single quantum well have been investigated using ~30 fs

17 pulses. The measurements on a 10 nm ZnSe QW shows marked quantum beats and pronounced non-exponential decay for delay smaller ~500 fs. Three-beam FWM experiments enabled the study of exciton formation processes from free electron-hole pairs and their effect on exciton dephasing. The investigation using sub-30 fs pulses on a

3 nm ZnSe SQW, where the exciton binding energy exceeds the LO-phonon energy, reveals a new exciton-LO phonon resonance at 31.6 meV above the exciton energy position.

An all optical photo-refractive effect (PRE) in ZnSe quantum wells resonant to the excitonic transition has been discovered. The effect is attributed to the formation of an electron grating within the QW formed by the interference of coherent excitons. Since the phase and the polarization of the incident pulses are preserved by the coherence of excitons, the PRE can be observed for pulses that do not overlap in time. The high diffraction efficiency of PRE makes it attractive for optical data processing where the angular momentum of incident photons provides an additional degree of information. In addition, PRE has potential for holographic optical coherent imaging (OCI) [2] for industrial and medical purposes.

The thesis has been organized in the following way: Chapter 2 describes the physical properties of ZnSe based semiconductors and quantum wells including band structure, excitons and their linear and non-linear optical properties. In addition, the effects of externally applied electric field on optical properties i.e. the electro-optical properties and the photorefractive effects are discussed. Chapter 3 explains coherent processes like the dephasing of excitons and the correlation of excitons with exciton as well as with phonons in semi-classical Boltzmann kinetic regime. It also contains a brief

18 introduction into the non-Markovian quantum kinetic regime. The construction of our home built tunable sub-20 fs Ti: Sapphire pulsed laser in the 820-885 nm range, the autocorrelation technique to measure the pulse width and the FWM experimental setup are described in chapter 4. Chapter 5 presents the experimental results that include the study of interband and interexciton coherences; carrier correlated dephasing of excitons, exciton-phonon interaction using 30 fs pulses, and investigations on the phase coherent photorefractive effect. The thesis closes with a summary in chapter 6 and references in chapter 7.

19 2. ZnSe based Semiconductor quantum wells

2.1. Electronic band structure

The electronic band structure plays a substantial role in the understanding of optical

properties in semiconductor crystals. The valence band electrons feel an effective

periodic lattice potential that is caused by the mean field of the nuclei and all other

electrons. The crystal periodicity can be described using primitive lattice vector

= + + R a1e1 a2e2 a3e3 , where ei defines fundamental translational vectors. The electronic

wave function in such effective potential is given by [1]

ψ = ⋅ −ω n ,k (r,t) un ,k (r) exp[ i(k r t)] )1(

= + + ( ) with wave vector k a1e1 a2e2 a3e3 , band index n and the Bloch function un,k r

( ) = ( + ) which has the full translational symmetry un ,k r un ,k r R . The energy Eigenvalues

= hω En (k) n (k) are known as the energy bands which provide the complete information of band structure of the crystal. The interesting region for optical transitions is near the center of the Brillouin Zone atk = 0 known as the Γ point, where the minimum of the conduction band occurs at the maximum of the valence band making ZnSe a direct gap semiconductor. In the Brillouin Zone, the photon wave vector q is very small compared

to the electronic wave numbers; hence in an absorption process the photon momentum

can be ignored. ZnSe crystals have zinc-blend Td symmetry; the valence band is p-type

= = 1 (orbital angular momentum l 1 and spin angular momentum s 2 ) which has a 6-fold degeneracy atk = 0. The spin-orbit coupling reduces this degeneracy and one of the three valence bands is shifted down in energy. This lower band is called split-off band

20 = + = 1 = ± 1 = ( J l s 2 , mJ 2 ). The top two bands which are still degenerate only at k 0

= 3 = ± 3 = 3 = ± 1 are known as heavy-hole ( J 2 , mJ 2 ) and light-hole band ( J 2 , mJ 2 ). The

= = 1 conduction band is s-like (l 0 and s 2 ) and is double degenerate due to spin. This is illustrated in figure 1.

E 1 1 J = ,m = ± 2 J 2

K = 3 = ± 3 J , mJ hh-band 3 1 2 2 J = , m = ± lh-band 2 J 2

The filling of the electronic energy states with electrons follows Pauli’s exclusion principle beginning with the lowest states. Absorption of a photon hω ( ≥ energy band

gap) takes an electron to the conduction band leaving behind a hole in the valence band.

The hole has the opposite charge of an electron. The effective mass of electron me and

hole mass mh are tensor quantities given by equation (2) where i, j are the coordinate indices.

1 1 ∂2E = e,h )2( h2 ∂ ∂ (me,h ) ji ki k j

Here Ee,h represents the electron and hole energy. For the isotropic parabolic conduction

band the energy is given by

21 h2 2 = k Ee (k) )3( 2me

= with me = 0.145 m0 [3]. Luttinger [4] showed that the (J 3 )2 -band with out considering the interaction with the (J = 1 )2 -band, can be described by the Hamiltonian

operator

h 2 k 2  5  h 2γ H = γ + γ  − 2 ()k 2 J 2 + k 2 J 2 + k 2 J 2 Luttinger 2m  1 2 2  m x x y y z z 0 0 )4( 2h 2γ − 3 ({ }{ }+ { }{ }+ { }{ }) k x k y J x J y k y k z J y J z k z k x J z J x m0

{ }= + Here ab (ab ba 2/) and the Ji represents the angular momentum matrix for

= = γ J 2/3 particles with i x, y, z . The Luttinger parameters i describe the parabolic

γ γ ≠ mean curvature ( 1) , the splitting of heavy and light-hole states ( 2 ) for k 0 and

γ = = γ = γ anisotropy of this splitting ( 3 ) . In spherical approximation (kx k y kz ) , then 2 3 , the hole bands are described according to

h2k 2 m 3 E (k) = z with m* = 0 for m = ± hh * hh γ − γ j mhh 1 2 2 2

h2k 2 m 1 E (k) = z with m* = 0 for m = ± lh * lh γ + γ j mlh 1 2 2 2

γ = γ = * = * = For ZnSe with 1 45.2 and 2 61.0 [5], mhh .0 813 m0 and mlh 27.0 m0 .

2.2. Excitons

The optical properties of semiconductors are significantly influenced by the presence of Coulomb interaction among the carriers. The Coulomb interaction between conduction band electrons and valence band holes causes the formation of bound electron-hole pairs

22 called excitons which can move freely through the crystal. Excitons in inorganic semiconductors considered in this thesis are called the Mott-Wannier excitons [6] where the electron and hole are weakly bound and the Bohr radius is large compared to the lattice constant. In this picture a bound electron-hole pair is similar to a neutral hydrogen atom where an electron is bound to a proton. In direct gap semiconductors, using effective mass approximation and parabolic bands the excitonic motion can be divided into the relative motion between electron and hole and the motion of their center of mass.

This leads to the energy dispersion relation

= + + EX Eg Er ER R = − y (Relative Motion) Er 2 )5( n h2K 2 (Center of mass) E = R 2M

Here n is the principal quantum number and Er provides the binding energy of the

exciton and Eg is the energy band gap. Ry is the exciton Rydberg energy given by

= ⋅ equation (6) with the exciton Bohr radius aB , the reduced mass mr (me mh M ) and

the background dielectric constant ε .

e4m 4πε ε h2 R = r ,a = 0 )6( y h2 πε ε 2 B 2 2 4( 0 ) mre

ε = = With 9 and mr 11.0 m0 [5], the value of Ry =19 meV and aB =4.3 nm. The total

= + energy of exciton involves the total wave vector K k e k h and total translational mass

= + M me mh providing the parabolic shape of the excitonic dispersion. The zero of the energy is the crystal ground state i.e. the crystal without any exciton. The excitonic wave function known as the Wannier wave function is given by

23 Ψ = φ i K⋅R (r, R) uc0 uv0 (r)e )7(

The first two terms of the Wannier wave function describe the characteristics of the atomic orbitals forming the conduction and the valence band respectively. In ZnSe the valence band is of p-type that forms the heavy-hole and light-hole band as explained in the band structure and the conduction electrons are of s-type. The exponential part describes the running wave character of excitons and φ(r ) is the exciton envelope function that explains the relative motion of electron and hole. The momentum conservation for excitonic optical transition K = q ≅ 0 leads to the vanishing center of

mass kinetic energy and the energy conservation determines that the created excitons

have the energy of incident photons. The angular momentum conservation is given

∆ = ± Γ = L byl 1 ( 5 excitons) and a series of sharp lines (n 1, 2, ) appear in the absorption

spectra. The dominant feature is the 1s-exciton absorption peak and the higher exciton

lines decreasing with 1 n3 hence appearing as small peaks just below the band gap.

With the recent development in crystal growth techniques it is possible to realize ultra-thin layers of a thickness comparable or smaller than the Bohr radius. In such structures the quantum mechanical wave-functions of carriers are confined inside the material. This gives rise to quantum confinement effects where the electron envelope wave-functions become quantized which leads to a substantial change in electronic and optical properties from the bulk material [1, 6]. In a quantum well structure the carriers like electrons and holes are confined in one space dimension (z) and are free to move in the plane perpendicular to the confinement directions (x and y). Such structure consists of

24 an ultra-thin layer of a semiconductor sandwiched between thin layers of a larger band gap semiconductor with matching lattice constant [1].

conduction band

E barrier well barrier

e E n =2 E z cb e

E n z=1

TEW B Egap gap Egap

Eh m z=1 Evb h

Em z=2

valence band z Figure 2: Schematic diagram of energy band structure of type-I quantum wells

In type I quantum wells the lowest electron and hole-states are confined in the well material. The energy difference between the larger band gap of the barrier and the smaller band gap of the well material provides the confinement potential for both electrons in the conduction band and holes in the valence band. In the type I quantum well band diagram

w B (see fig. 2) Evb and Ecb are the valence and conduction band offset, Egap and Egap are the band gap energy of well and barrier, respectively. The energy Eigenvalues for an electron or hole confined in one dimension in the quantum well with infinite potential are [1]

h2 2π 2h2 = + ()2 + 2 = j = E E j k x k y , E j 2 , j 3,2,1 ... )8( 2me,h 2me,h Lz

Here j is the sub-band quantum number and Lz is the thickness of the well in z-

direction. me,h is the mass of electron or hole. In quantum wells the total energy of the bound excitons become

25 R h2 j2π 2h2 E 2d = E 2D − y + ()K 2 + K 2 ,n = 3,2,1 ..... , E 2D = E + )9( X g − 2 x y j g g 2 (n j 1 )2 2M 2mr Lz

− −3 In this case the oscillator strength is proportional to (n j 1 )2 and the Bohr radius is

2D ∝ − aB aB (n j 1 )2 . The reduction to two dimensions leads to a reduction of the exciton

Bohr radius. Due to the stronger overlap between of electron and hole the exciton binding energy increases by a factor of 4 compared to bulk value.

V(z)

I III II

− 0 z LZ 2 LZ 2

Figure 3: Schematic diagram of a finite potential well of depth V0 given by equation (10) after [1].

The above described case is an ideal 2d system with infinite barrier and can not be realized in practice, so it is required to consider a finite confinement potential well (see

0 for − L 2 < z < − L 2 fig. 3) V (z) = z z 10( )  > V0 for z Lz 2

The Schrödinger equation for the x-y motion is unchanged comparing to the infinite barrier but the equation for z-motion has to be solved separately for three different regions I, II and III (see fig. 3). By matching the wave-functions and their derivatives at

± the interfaces Lz 2 , the transcendental equation for energy eigen values for even states is given by

26  +  + mE + E tan  z L  = V − E 11( ) z  2 z  0 z  2h 

and for odd states,

 −  − mE − − E cot  z L  = V − E 12( ) z  2 z  0 z  2h 

The roots of the transcendental equations (11) and (12) have to be determined numerically. The number of bound states in the potential well depends on the depth of the

well V0 . As long as V0 is positive, there is always at least one bound even state, the

ground state. For more than one quantum confined states, the symmetry between the

successive higher states alternates, until the highest bound state comes. In region II, the

wave-functions become oscillatory with in the well but penetrate through the barrier into

regions I and III where they decay exponentially. Absorption

Figure 4: Schematic absorption spectra of a two dimensional semiconductor which shows each step of the step function acquires its own excitonic series, after [1].

For finite barriers and with well width Lz the exciton binding energy can be estimated using the model of Mathieu et al [7]. The calculation introduces a fractional- dimensional space with dimension α . In the limit of a bulk semiconductor and the ideal two-dimensional case the α values are α = 3 and α = 2 , respectively. For real quantum

27 wells α lies between limiting values 2 and 3 [8]. Due to the penetration of the wavefunction into the barrier the binding energy does not reach the maximum value

2D = EB 4Ry . In two-dimension the density of states in QWs becomes step function [see

equation (13)]. The steps occur at the quantized energy values E j with their own excitonic series (see fig. 4).

m S 2d (E) ∝ Θ(E −E ) 13( ) π 2 ∑ j h j

Electron states

= 1 = + 1 = 1 = − 1 J 2 , mJ 2 J 2 , mJ 2

σ + σ −

= 3 = + 3 = 3 = − 3 J 2 , mJ 2 J 2 , mJ 2

= 3 = − 1 = 3 = + 1 J 2 , mJ 2 J 2 , mJ 2 Hole states

Figure 5: The diagram depicts the allowed optical transitions for the HH and LH excitons, after [9].

The excitonic transitions occur between the quantized electron and hole states with the same quantum number j i.e. ∆j = 0 . The excitonic 1s transition is well resolved and the magnitude of higher exciton transition lines varies with j like ( j −1 )2 −3 . The most striking feature of excitons in quasi-two-dimensional systems is the splitting into light- hole (LH) and heavy-hole (HH) excitons which results from the corresponding splitting of the valence band at the Γ point. Figure 5 depicts the allowed optical transitions for HH

28 ∆ = + ∆ = − and LH excitons. The excitons with m j 1 and m j 1 can be excited by the photons of spin σ + and σ − respectively.

2.3. Linear and non-linear optical properties

= −i ω t A monochromatic field E(t) E0e can create an electric polarization in a

dielectric medium like semiconductors and it is defined as the number of dipoles per unit

volume. The value of the first order polarization is given by [10]

(1) = χ )1( − P (t) ∫ (t t1)E(t1) dt 1 14( )

After Fourier transformation into frequency domain the polarization can be written as

P(ω) = χ(ω)E(ω ) 15( )

Equation (15) expresses the linear dependence of the polarization P(ω ) on the field

E(ω) through the electric susceptibilityχ(ω ) which is a tensor quantity. With electric

= ε + displacement vector D 0E P , the dielectric function of the medium can be expressed asε (ω) = 1+ χ(ω ) . This further provides the linear optical properties such as the refractive index Re 1+ χ(ω) and the absorption coefficient of a semiconductor

α(ω) ∝ Im [ε (ω)].

At sufficiently high excitation intensities, semiconductors exhibit nonlinear t optical properties. The susceptibility tensor χ(ω, E ) which relates different vector

component of polarization Pi and the electric field Ei can be expa nded as a power series

= χ (1) + χ (2) ⋅ + χ (3) ⋅ ⋅ +L PNL (r,t) E E E E E E 16( )

29 The susceptibility coefficientsχ )1( , χ )2( , χ )3( ... are tensors. The first term in the above equation is linear in the field E and the first order susceptibility χ (1) which describes the linear response of the medium as described in equation (15). The second term containing the second-order susceptibility χ )2( describes effects like second harmonic, sum- and

difference-frequency generation. For crystals with inversion symmetry, χ )2( and all other

even terms vanish. The third order susceptibility χ (3) is responsible for the generation of

four-wave mixing, (and other nonlinearities like hyper-Raman scattering, coherent anti-

Stokes Raman scattering etc). The third order polarization is given by [10]

( ) P 3 r( ,t ) = 17( ) χ )3( − − − − − − ∫ r( 1 r,r 2 r,r r ,3 t t1,t t2 ,t t3 ) 1,r(E t1 2 ,r(E) t2 3,r(E) t3 )dr1 dt 1dr2 dt 2dr3 dt 3

In the general case the applied field consists of a superposition of monochromatic waves,

i.e.

= ⋅ − ω + E(r,t) ∑E0i exp[ i(k i r it) c c.]. 18( ) i

Using this equation and after Fourier transformation into the frequency domain the third

order nonlinear polarization can be written as

(3) ω = χ )3( ω ⋅ ω ω ω P (k, ) (k, ) E1(k1, 1)E2 (k 2, 2 )E3 (k 3, 3 ) 19( )

χ )3( − − − − − − The Fourier transform of the response function (r r1,r r2 ,r r ,3 t t1,t t2 ,t t3 ) is

χ )3( ω = ± ± ± ω = ±ω ± ω ± ω (k, ) , where k k1 k 2 k 3 and 1 2 3 describe the momentum and

energy conservation, respectively.

30 2.4. Electro-optical properties and photorefractive effect

Franz and Keldysh studied the optical absorption as a function of an externally applied electric field in semiconductors [1, 6]. Without considering Coulomb interaction the

e = − ⋅ interaction Hamiltonian of this electro-absorption is Hint eE0 re where E0 is the

applied electric field strength. Since the center of mass of an electron-hole pair has no net

charge, it is not affected by the electric field. The relative-coordinate terms, however,

enter the Schrödinger equation. This so called Franz-Keldysh effect leads to the appearance of absorption below the band gap and to oscillatory absorption variations for energies above the band gap. The inclusion of Coulomb interaction between electrons and holes in bulk semiconductor describes the exciton absorption and the effect of electric field is known as DC Stark effect . In this case the interaction Hamiltonian becomes

e2 H e = − − eE ⋅ r 20( ) int πε 0 e 4 0r

The modified Coulomb well leads to a broadening of the exciton resonances as well as to shift to lower energy [1].

In quantum wells the electric field can be applied either parallel or perpendicular to the QW layers. For parallel applied electric field the motion of the carriers is not affected by the confinement and the effect is similar to the bulk semiconductor. The treatment of a perpendicular electric field taking the Coulomb interaction into account is referred as

Quantum Confined Stark Effect (QCSE) [11, 12, 13, 14]. An electric field perpendicular to the QW layers can tilt the conduction and valence band hence modifying the overlap between electron and hole wave functions. This leads to a red shift of exciton

31 position and to a broadening of the exciton resonance. The higher binding energy in QW compared to bulk semiconductor enables the excitons to persist up to high electric field magnitudes without ionization.

Interference fringes Light

Gold contacts

Buffer Parallel Geometry Field MQW Lines : Franz-Keldysh Effect Buffer

Field MQW Perpendicular Geometry Lines : QC Stark Effect

Figure 6: Parallel and perpendicular geometries of a photorefractive quantum well structure, after [15].

Quantum-confined excitons as well as high carrier mobilities make semiconductor quantum wells a very sensitive material for photorefractive phenomena that are based on the electro-optic effects [15]. In quantum wells, excitons show a large Franz-Keldysh effect when the electric field is applied parallel to the QW layers [16, 17, 18, 19, 20, 21] or high quantum confined Stark effect (QCSE) for the electric field normal to the QW layers.

The two different geometries of photorefractive effects in multiple QWs are shown in fig. 6. The intentionally doped QWs are illuminated by the interference of two laser beams providing bright and dark fringes on the surface. The rate of generation of charged carriers is locally a maximum at the position of maximum intensity. The external electric

32 field is applied using gold contacts. In parallel geometry the direction of the electric field is parallel to the growth direction of the quantum well while in the perpendicular geometry the electric field is perpendicular to the growth direction as shown in the fig. 6.

The externally applied electric field can drift the charge carriers to the borders of dark fringes and the charge carriers can be trapped there. The trapping of charge carriers can be made possible by the high density of impurities. The trapped carriers will not be re- excited provided they are in the dark regions. This leads to a modulation of space charge distribution and hence modulation of electric fields in side the quantum well. The resulting diffraction efficiency η is defined as the ratio of the diffracted intensity relative to the incident intensity and the typical values lie between 10 −5 and 10 −3 .

33 3. Dephasing of excitons : Theoretical background

3.1. Dephasing of excitons: Boltzmann regime

When a semiconductor is excited by a short laser pulse with energy resonant to the exciton transition energy, the excitons oscillate coherently with the light field forming a macroscopic coherent polarization P . The quantum mechanical phase of the polarization is defined by the driving laser field. The coherent regime starts at this point. The coherence can not be maintained for arbitrarily long time because scattering events within the system or interactions within the surrounding environment destroy the phase relations. The processes which destroy the phase include recombination, collision with phonons, collision with excitons and with other excitations. The typical time scale for such processes in semiconductors is in the range of picoseconds or shorter. After a time

t , the part of the polarization which is still coherent with the exciting pulse can be written to a first approximation as

= − t  Pcoh P0 exp   21( )  T2 

Here T2 is referred to as the dephasing time of excitons. The coherence of the excitonic polarization is ultimately destroyed by the decay of the exciton population in

recombination processes which can be described by a life time T1 . This decay of exciton population can be written as

= − ∝ 2 ∝ − 2 N N0 exp[ t T1] P P0 exp[ t 2( T1 ]) 22( )

From equation (22), the hierarchy of time constants for a two-level system is given by

* ≤ equation (23) with T as the pure dephasing time and T2 2T1

34 = + * 1 T2 1 2T1 1 T 23( )

3.1.1. Principle of four-wave mixing (FWM)

A versatile technique to study the coherent regime is four-wave mixing (FWM).

In this experiment, the material is excited with two or three beams which may or may not

have same frequencies and are incident under certain angles and the generation of the

FWM signal can be described by the third-order nonlinear polarization given in equation

(19). If the frequencies of the beams are identical the mixing process is called degenerate

four-wave mixing (DFWM). The loss of the phase coherence can be directly monitored in

the time domain by transient FWM experiment using ultra-short laser pulses.

ω A pulse at time t1 with wave vector k1 and energy 1 being resonant with the excitonic transition energy creates a coherent exciton polarization. The coherently excited quasi-particles loose their phase due to different scattering processes like exciton-exciton

scattering and exciton-phonon scattering. A second pulse at time t2 with wave vector k 2 ,

τ = − that is delayed by a time 12 (t2 t1) (shorter than the dephasing time of excitons), also

creates a coherent polarization of excitons that interferes with the non-dephased part of

the coherent polarization created by k1 . This interference leads to the formation of a

± − spatially periodic exciton grating with the grating wave vector (k 2 k1) and the grating

Λ = π − τ constant 2 k 2 k1 . The third beam of wave vector k 3 , delayed by a time 23

with respect to the second pulse, sets up the third-order nonlinear polarization. This can

± − be diffracted by the formed grating into the phase matching direction k 3 (k 2 k1 )

leading to the fourth coherent output beam in both transmission and reflection geometry.

This describes the FWM process which involves three incident beams. In two-beam

35 FWM experiments, which involves only k1 and k 2 , the delayed beam k 2 can be self

± − − diffracted from the grating (k 2 k1) into the phase matching direction of 2k 2 k1 and

− 2k1 k 2 via third order nonlinear process. In case of DFWM experiment all of the three incident beams are from the same laser source having constant frequency. Figure 7 illustrates a standard schematic diagram for three-beam DFWM experiment.

FWM-signal + − k3 k 2 k1 k 1

k 3

k 2

Figure 7: Schematic diagram of a three-beam FWM experiment.

= Perfect phase matching condition in two-beam degenerate FWM occurs at k k1 and

= k k 2 which are the directions of the incident beams. As in the case with only one beam,

the nonlinear signals are generated in a direction with a large linear signal and may

therefore be difficult to detect at moderate intensities [22]. There are two near-phase

= − − ∆ = − − ∆ ∆ matching conditions k 2k 2 k1 k T and k 2k1 k 2 k T , here k T is the wave

∆ vector mismatch perpendicular to the sample plane in transmission direction and k R is

the wave vector mismatch in reflection direction as shown in figure 8. The advantage in

this geometry is that the nonlinear signal is generated in a background free direction. The

linear background can be eliminated using simple spatial filters.

36 Sample

Figure 8: Diagram showing the phase mismatch and the phase matching direction in a 2-beam FWM experiment.

The signal intensity in the scattered phase matching direction is given below with d as the sample thickness [23].

sin 2 (∆k ⋅ d )2/ I α d 2 T 24( ) s ∆ ⋅ 2 ( k T d )2/

∆ ⋅ ≤ Near phase matching condition requires k T d 1 and is well satisfied in very thin samples like QWs. Hence in QWs, almost the same intensity can be observed in the back scattering direction as in the forward direction [22]. The same condition is also necessary to satisfy the phase matching condition for three-beam FWM experiment.

The FWM experiment not only enables the investigation of the relaxation times of excitons but also make it possible to provide information about the coherent dynamics of carriers. Intensity dependent measurements can provide exciton-exciton as well as exciton-electron-hole scattering parameters [24, 25, 26, 27, 28, 29]. FWM signals at different energetic positions can be identified by quantum beat spectroscopy [30, 31, 32] and polarization dependent experiments. The interaction dynamics of exciton-phonon

37 scattering can be obtained using temperature dependent FWM measurements [29, 33, 34].

Furthermore the three-beam experiment provides more degrees freedom. The three-beam experiment enables to determine the lifetime of excitons (more accurate the combination of grating diffusion and exciton recombination) by probing the induced grating by the third beam independently that does not require coherence to the first two beams.

3.1.2. OBE for homogeneously broadened system

A general theoretical approach to analyze the coherent optical phenomena is

based on the semiconductor Bloch equations (SBE) [35] which takes into account the fermionic nature and the Coulomb interaction of carriers in a semiconductor. However, under certain excitation conditions the important mechanisms can be obtained by simplified phenomenological multilevel models, essentially representing extended optical

Bloch equations (OBEs). These models consider e.g., local-field effects (LFE) [36]

excitation-induced dephasing (EID) [37, 38, 39] and biexciton formation (BIF) [40, 41,

42]. Here the extended OBE for a two-level system has been described and the

calculation has been extended to a ten–level system.

The optical Bloch equations derived for an ensemble of independent two-level

systems are generally used to describe the coherent phenomena in atoms and molecules

[43]. In the simplest approximation resonantly created excitons in semiconductor can be

considered as an ensemble of two-level systems consisting of the crystal ground state and

an excited state [9]. In semiconductors excited electrons do not live in pure states but

occur as mixed states which can be described through the density matrix formalism. In

this formalism, considering the homogeneously broadened system and for degenerate

38 − two-pulse FWM experiment the third order polarization diffracted into 2k 2 k1 direction is given by

(3 ) 2N i * * * P − ()t,τ = − ()() ⋅ ε ⋅ ε ( ⋅ ε ) 2k 2 k1 12 h3 21 21 2 21 2 21 1 25( ) Θ −τ Θ τ []Ω τ − []Ω* τ (t 12 ) ( 12 ) exp i 21 ( 12 t) exp i 21 12

After Fourier transformation

ωτ (3 ) 2N i * * * * exp[ i ] P − ()ω,τ = ()() ⋅ε ⋅ε ( ⋅ε )Θ(τ ) exp [i Ω τ ] 26( ) 2k 2 k1 12 h3 21 21 2 21 2 21 1 12 21 12 ()ω − Ω 21

Θ τ The Heaviside function ( 12 ) indicates that the signal is generated only for positive

τ ≥ τ < delay time i.e. for 12 0 and the FWM signal is zero for 12 0 . It indicates that when

the electric field of beam k 2 comes first it does not find an exciton density grating and

− therefore does not result any signal in direction 2k 2 k1 .

The intensity of the observed signal is

2 [− τ γ ] )3( (3 ) exp 2 12 21 I (ω,τ ) = P − ()ω,τ ∝ 27( ) 21 12 2k 2 k1 12 ()()ω − ω 2 + γ 2 21 21

The signal intensity has a Lorenztian line shape and the signal decays with a time rate

γ = of 2 21 2 T2 . This analysis is appropriate for homogenous broadened systems with

Γ = h − spectral line width 2 T2 . The emitted FWM signal in the 2k 2 k1 direction for

homogeneously broadened system of excitons is called free polarization decay (FPD),

starting in real time with the arrival of the second pulse k 2 .

3.1.3. OBE for an inhomogenously broadened system

The dephasing time of every excited exciton determines the spectral width of the emission line. In the case of homogeneous broadening each exciton has the same

39 transition energy and spectral width. The exponential decay of the exciton polarization leads to a Lorentzian shape of the spectral line. Because of surface roughness, strain fields, and impurities different excitons may oscillate with different resonant frequencies and the actual emission is the superposition of a large number of Lorentzian lines. The total emission is said to be inhomogeneously broadened and the normalized inhomogeneous function has a Gaussian line shapeg(ω ) , where g(ω) dω defines the

number of excitons with resonance center frequency ω in interval dω . In case of an inhomogeneously broadened system the result obtained from the optical Bloch equation has to be integrated with the inhomogeneous broadened Gaussian distribution given by

2 ln4 2   ω − ω   g (ω) = exp − 4× ln 2×  21   28( ) N Γ π   Γ     Inh  

Γ The inhomogeneous broadening has the full width at half maximum (FWHM) of Inh .

The polarization in real time t is given by is found to be [44]

 Γ2 − τ 2  )3( )3( )3( Inh (t 2 12 ) P − ()()()t,τ = g (ω) P − t,τ ,ω dω = P − t ⋅ exp −  29( ) 2k 2 k1 12 Inh ∫ N 2k 2 k1 12 2k 2 k1    16 ln 2 

For an inhomogeneously broadened system, the phases of different frequency components in the spectral bandwidth of the laser evolve at different rates. Due to destructive interference, the macroscopic polarization of the ensemble decrease with a

Γ time constant inversely proportional to Inh . When the second pulse arrives at delay time

τ 12 (smaller than T2 ), it reverses the phase evolution of the different frequency components. This leads to the generation of FWM signal and the process is known as

Photon Echo [6, 43]. Equation (29) indicates that the photon echo emitted at t = 2τ is the real time Fourier transform of the inhomogeneous line and the Θ -function clips the

40 Gaussian in real time for delays smaller than the inverse line-width τ < 1 Γ [44]. The

− τ τ > Γ FWM signal intensity is proportional to exp( 4 12 /T2 ) for 5 Inh . Accordingly the

decay time constant is T2 4 which by factor of two smaller than for homogeneous

broadened system. Fourier transformation of equation (29) leads to the following third-

order polarization for inhomogeneous broadening.

)3( P − (ω,τ ) 2k 2 k1 12 Inh 2 2 4 π ln 2 ωτ − γ τ  (ω − ω) − γ  = Θ()()τ ⋅ 4i N µ e2i 12 e 2 21 12 exp − ln4 2 21 21  30( ) 12 21 Γ  Γ2  Inh  Inh   γ (ω − ω)    γ Γ ⋅τ (ω − ω)  × exp i ln8 2 21 21  ⋅ erfc 2 ln 2 21 − Inh 12 + i 21   Γ2   Γ Γ   Inh    Inh ln8 2 Inh 

3.1.4. Various FWM processes

At low excitation densities and at low temperature the FWM signal at excitonic

transitions in semiconductors can be explained using OBEs. This nonlinear third-order

signal is generated due to the diffraction of the probe beam by the spatially periodic

optically-induced exciton density (population) grating according to the phase-space-

filling (PSF). The occurrence of FWM signal at negative delay and in various

polarization configurations require several modification of OBEs by phenomenological

extensions like excitation induced dephasing (EID) [37, 38, 39], biexciton formation

(BIF) [40, 41, 42], and less important in II-VI semiconductors by local field effect (LFE)

[36].

Phase space filling (PSF)

Pauli’s Exclusion Principle forbids fermions to occupy quantum states more than one at a time. Therefore in energy bands, every k state can be occupied twice with spin

41 = − − up and spin down. The band filling factor A(k) 1 fe (k) fh (k ) determines the

availability of an electron and a hole state at k withfe (k ) and fh (k ) being the Fermi functions of electrons and holes, respectively. The filling of the bands leads to a gradual bleaching of the absorption close to the band gap energy and results in a shift of the absorption to higher energies (blue shift). This change in absorption leads to an optical nonlinearity. The influence of A(k ) on optical processes is called the phase-space-filling

[1].

Excitation Induced Dephasing (EID)

The density dependence of FWM signal is phenomenologically described by excitation induced dephasing (EID). At low excitation limit the Taylor expansion of the dephasing rate γ in terms of exciton density is given by following equation [45].

γ = γ 0 + γ ′ ⋅δ = γ 0 + σ ⋅ (nX ) (nX ) nX (nX ) nX 31( )

σ 0 = γ 0 The EID parameter is and nX 0 is the background exciton density and (nX ) is the

dephasing rate at zero exciton density. Accordingly EID considers that the dephasing rate

increases with increasing exciton density nX due to a stronger exciton-exciton ( X-X) scattering. At very high densities the dephasing rate saturates due to exciton bleaching and the EID contribution diminishes.

= ~ Θ − EID modifies the OBEs through the exciton density N1 N1 (t t1 ) created by

= ~ Θ − pulse 1 and the exciton density N2 N2 (t t1 ) created by pulse 2. The term

Ω = ω − Ω~ = ω − − σ + 21 21 i T2 in equation (42) is modified to 21 21 i T2 i (N1 N2 ) . EID also

42 leads to the diffraction of first order exciton polarization. So the third order equation is

σ ⋅ ρ )2( ρ )1( modified by the term 22 21 and it leads to

~ 2i ρ& )3( + iΩ ρ )3( = − ⋅ E(r,t)ρ )2( − N σ ⋅ ρ )2( ρ )1( 32( ) 21 21 21 h 21 22 22 21

− The macroscopic third order polarization for FWM signal into the direction of 2k 2 k1 can be obtained as

(3 ) N * * * P − ()ω,τ > 0 = ()() ⋅ε ⋅ε () ⋅ε Θ(t −τ )Θ(τ ) 2k 2 k1 12 3 21 21 2 21 2 21 1 12 12 h 33( )  σ  × []Ω* + σ τ ωτ ⋅ 2 + i N exp i( 21 i N1 ) 12 exp[ i 12 ]  ~  ()ω − Ω ()ω − Ω~ 2   21 21 

(3 ) P − ()ω,τ < 0 34( ) 2k 2 k1 12 N  iσ N  = * ()() ⋅ε ⋅ε ()* ⋅ε* Θ(t)Θ(−τ ) exp []2i( Ω − iσN )τ ⋅  h3 21 21 2 21 2 21 1 12 21 2 12 ()ω − Ω~ 2  21 

Here N is the total number of available exciton (two-level systems) density. The signal for negative delay due to EID decays two times faster than the signal for positive delay

σ + and the spectral FWM line-shape is broadened due to the term (N1 N2 ) . EID can

provide the explanation for the high ratio of the time integrated FWM intensities for

collinear ( ↑, ↑) to cross-linear ( ↑, →). Since the σ + and σ − gratings in ( ↑, →)

configuration are out of phase the (EID) contribution vanishes whereas the gratings are in

phase for ( ↑, ↑) configuration which leads to a high EID signal in this configuration. This

indicates that EID plays an important role in ( ↑, ↑) configuration.

Biexcitonic Effects (BIF)

When two excitons are close together, the two exciton wave-function can be

either a symmetric or an anti-symmetric combination of the exciton wavefunctions. The

43 symmetric combination has lower energy and forms the bound state of excitonic molecule referred as biexciton ( XX ).

5 4

XX h

2 3 X h X h

+ = σ σ − = − mJ 1 mJ 1

Figure 9: The model shows the formation of the biexciton through two opposite circularly polarized light pulses.

The heavy-hole exciton X h involves a transition from the heavy-hole sub-band

( = 3 = ± 3 ) with angular momentum Eigenstates J 2 , mJ 2 to the electron sub-band

( = 1 = ± 1 ) ( = = ± ) with J 2 , mJ 2 , respectively. So the X h Eigenstates J ,1 mJ 1 are excited

σ ± ± = − − 5.0 µ (m ) by circularly polarized light with the dipole matrix vectors h 2 h ,1 i in

Jones vector notation. The biexcitons are created by a two photon coherent excitation

schematically shown in the energy level diagram fig 9. The bound heavy-hole biexcitons

XX h have pairs of opposite electron as well as hole spins as a consequence of the Pauli exclusion principle [46].

The two-photon absorption process leads to two-photon coherence (TPC) [9]

between the ground state and the biexcitonic state. According to TPC the subsequent

pulse can be emitted into the direction satisfying the phase matching condition. The

XX h TPC induced FWM signal can be observed for positive and negative delay. At

44 negative delay the signal is created by the simultaneous σ + and σ − excitation of the

pulse k 2 before the pulse k1 arrives at the sample. The third order polarization

(3 ) P − (ω,τ ) generated by pulse k gives rise to a time integrated FWM signal 2k 2 k1 12 1

(3 ) I − (ω,τ ) at the X transition at energy hω and at the biexciton induced transition 2k 2 k1 12 h h

hω − ∆ ∆ at energy h h , where h is the heavy-hole biexciton binding energy. Since the

bound biexcitons are formed with excitons of opposite spin the biexciton induced FWM

signal can be suppressed by using circularly polarized light.

3.1.5. Exciton-exciton scattering

The exciton-exciton scattering rate of mainly coherent excitons can be studied in two-beam FWM experiments by changing the intensity of the excitation pulses [47]. At not too high excitation intensities the variation of the determined homogeneous line-

Γ = hγ width Hom (nX ) 2 (nX ) as a function of the exciton density shows a linear dependence.

The exciton density per unit area nX can be determined from the absorption measurement using the formula [48]

N 1( − e−OD )( 1− R)w n = hν 35( ) X π a2d

Here Nhν is the total number of photon in the pulse, w is the ratio of the X h absorption

line-width to the spectral width of the laser pulse, and R accounts for reflection losses.

OD is the peak optical density of the X h transition, d is the ZnSe layer thickness, and a is the 1 e2 focus diameter of the pulses on the sample.

45 3.1.6. Exciton-LO-phonon interaction

Interactions of excitons with phonons in a semiconductor play a significant role in dephasing processes of excitons. This exciton-phonon interaction can be understood using the deformation potential and piezo-electric effect in semiconductors. In the long- wavelength limit of acoustic phonons, the atomic displacements correspond to the deformation of the crystal. Such a deformation will change electronic energies induced by the static distortion of the lattice known as deformation potential [49]. Optical phonons can be regarded as microscopic distortions with in a primitive unit cell. In polar crystals a long-wavelength longitudinal optical (LO) phonon results due to uniform displacement of the charged atoms with in the primitive cell. Such relative displacement of oppositely charged atom generates a macroscopic electric field [50, 51].

The effect of acoustic and longitudinal optical phonon interaction on the exciton linewidth can be determined using the following relation expected in first order perturbation theory [52, 53],

β γ (T,n ) = γ ,0( n ) + β T + LO 36( ) hom X hom X AC − exp( ELO kBT ) 1

Here ELO is the energy of LO phonon, for ZnSe ELO =31.6 meV. kB is the Boltzmann

γ β constant, hom ,0( nX ) contains the low-temperature homogeneous line-width. LO and

β AC are respectively the exciton-LO phonon and exciton- acoustic phonon scattering parameter which can be determined using the temperature dependent two-beam four- wave mixing experiment.

46 3.2. Extended OBEs: Ten level model

The two-level system is unable to provide detail understanding of the FWM signal

obtained from different excitonic states like heavy-hole, light-hole exciton and biexcitons

etc. So the third order Optical Bloch equations in the external fields for δ-shaped laser

pulses have to be extended to higher-level model as well as taking consideration of

phenomenological terms like EID, BIF.

| 10> | 8> | 9>

| 5>

| 4>

| 6> | 7>

| 2>

σ+ σ−

| 1>

Figure 10: Ten-level model.

In this thesis a 10-level model (see fig. 10) has been presented with the ground level 1

+ − initially populated takes into account the interaction of σ and σ Xh heavy-hole (levels

2 and 3 ) and Xl light-hole excitons (levels 6 and 7 ) and also allows for the

formation of bound heavy-hole biexcitons XX h and biexciton continuum (levels 4 and

5 ). Furthermore the formation of mixed biexcitons XX m (levels 8 and 9 ) as well as of a light-hole biexciton continuum (level 10 ) is introduced into this model. The

corresponding OBE are further extended to encompass EID, as discussed in Ref. (39).

47 Local field effects (LFE) have not been considered since they are less important in ZnSe quantum wells as shown in previous two-beam FWM experiments [42]. After Fourier

+ − transformation the third-order polarization into k 3 k 2 k 1 direction for negative

(τ < )0 and positive (τ > )0 delay reads:

P )3( (ω ,τ < )0 = k 3 + k 2 −k1 N * Θ (−τ ) [ sg [( ⋅ e )( * ⋅ e * )( ⋅ e ) exp( iΩ τ ) h 3 ∑ ()ω − Ω~ s ' g 2 s ' g 1 sg 3 sg s , s ,' t sg + * ⋅ * ⋅ + ⋅ * ⋅ * ⋅ Ω τ [( st e1 )( st ' e 2 ) ( sg e 2 )( s ' g e1 )]( s ' g e 3 ) exp( i s ' g )] 2 * − st ( ⋅ e )( * ⋅ e * )( ⋅ e ) exp( iΩ τ ) 37( ) ()ω − Ω~ st ' 2 sg 1 s ' g 3 s ' g st * + iN β sg (2[ ⋅ e )( ⋅ e )( * ⋅ e* ) exp( iΩ τ ) ()ω − Ω~ 2 sg 3 s ' g 2 s ' g 1 sg sg + ⋅ ⋅ * ⋅ * Ω τ ( s ' g e 3 )( sg e 2 )( s ' g e1 ) exp( i s ' g )] ]

P )3( (ω ,τ > )0 = k 3 + k 2 −k1 N 2 *  τ  Θ (τ ) exp( iω τ ) [ sg ([ ⋅ e )( ⋅ e )( * ⋅ e * ) exp  −  h 3 ∑ ()ω − Ω~ sg 3 s ' g 2 s ' g 1   s , s ,' t sg  Tc  + ( ⋅ e )( * ⋅ e * )( ⋅ e ) exp( −iΩ τ ]) s ' g 3 s ' g 1 sg 2 ss ' 2 * − st ( ⋅ e )( ⋅ e )( * ⋅ e * ) exp( −iΩ τ ) 38( ) ()ω − Ω~ st ' 3 s ' g 2 sg 1 s 's st  τ  * + i Nβ exp  −  sg [ (2 ⋅ e )( ⋅ e )( * ⋅ e * )  T  ()ω − Ω~ 2 sg 3 s ' g 2 s ' g 1  c  sg + − Ω τ ⋅ ⋅ * ⋅ * Ω * τ exp( i sg )( sg e 2 )( s ' g e 3 )( s ' g e1 ) exp( i s ' g ]])

σ + σ − In equations (37) and (38), e j denotes the polarization unit vectors of the and polarized laser pulses with σ± = 2− 5.0 ,1( mi) in Jones notation and j = 1, 2, 3

corresponding to the three laser pulses. The indices s and s ' correspond to the single

exciton states ( 2 , 3 , 6 , 7 ) whereas t refers to the two-exciton states ( 4 , 5 ,

48 8 , 9 , 10 , g being the ground state 1 . The optical matrix elements from ground

= µ + µ − µ state g to states s are gs where gs gs σ or gs σ where gs accounts for the

magnitude of the dipole transitions. Furthermore we have used the following

abbreviations:

Ω = ω − γ = ω − (γ 0 + β ) sg sg i sg sg i sg sg N3 Ω = ω − iγ = ω − i()γ 0 + β ()N + N s's s's s's s's s's s's 1 2 Ω~ = Ω − β + sg sg i sg (N1 N2 ) 39( ) Ω~ = ω − ()γ 0 + β + + ts ts i ts ts (N1 N2 N3 )

γ γ = −1 The dephasing rates sg are given by sg (Tsg ) , where Tsg are the interband

dephasing times of different Xh and Xl exciton to ground transitions. For s ≠ s ' the

γ γ = −1 interexciton dephasing rate s's is given by s's (Ts's ) . The time constant Ts's describes the loss of coherence between two Xh, two Xl and Xh-Xl exciton states with different spin ( T32 ,T76 ,T63 ,T72 , respectively) as well as between Xh-Xl exciton states

= with equal spin ( T62 ,T73 ). For s s ' the decay rate of the exciton population grating is

γ = −1 given by s's (Tc ) . Tc is a combined time constant taking into account the exciton lifetime as well as the exciton diffusion inside the population grating. The

ω ω ω quantities sg ,,s's ts are the angular frequency between states g, s ,'s and t, N is the

γ 0 γ 0 total density of the exciton system. sg and s's denote the dephasing rates at zero

β β β exciton density and sg , s's , ts are the exciton-exciton scattering parameters taking into account an increased dephasing rate of various transitions with exciton densities

N1, N2 and N3 created by the three-laser pulses k1 ,k 2 and k 3 , respectively. The

49 β = β exciton-exciton scattering parameter sg is also responsible for the EID terms in equations (37) and (38).

3.3. Brief introduction into quantum kinetics

The ultra-fast dynamics of semiconductors is dominated by many particle correlations.

The strongest effects are mediated by Coulomb interaction but other couplings e.g. to lattice vibrations also play a significant role. Semi-classical theory treats interaction processes as scattering events that take place at a certain point of time and space and usually they are explained by a theory called Boltzmann kinetic theory. This theory follows the Fermi’s golden rule where scattering processes strictly conserve energy and momentum and the energy conservation holds only if the carrier distribution function varies slowly on the scale of collision time [54]. Since under these assumptions individual collision events become decoupled, the memory of what has happened prior to the collision disappeared. This is known as the Markovian approximation [9] under

which most of the results of coherent spectroscopy have been analyzed in the OBEs. The

Markovian approximation can be considered valid if the average interval between

collisions (inverse of the scattering rate) is much longer than the collision duration.

Quantum kinetic theory [55, 56] accounts for a finite duration of interactions and thus

allows for a temporal resolution of dynamics of the scattering process and in this case the

dephasing time depends on the “history” of the environment. Quantum kinetic

phenomena are therefore often also called memory effects [9, 57] in the non-Markovian

approximation. In this regime the wave-like nature of interacting quasi-particles leads to

mixed state coherences [57].

50 Recent progress in experimental techniques as well as in quantum kinetics theories have made it possible to study such quantum coherent phenomena using 10 fs optical pulses. The first experiments on electron-phonon quantum kinetics were published several years ago on bulk GaAs (see fig. 11) [58].

Figure 11: TI-FWM from GaAs at 77 K using 14.2 fs pulses for three different carrier densities that shows the beating between inter-band polarizations connected by LO phonon scatterings, from [58].

In FWM experiments with 14.2 fs pulses, the FWM trace at the exciton energy reveals oscillations, which were interpreted as a beating of intraband-polarization components with frequencies ω and ω′ connected by coherent LO-phonon scattering of an electron in the conduction band. These frequency components are resonant with the band states k and k′ . The two interfering momentum states are coupled by an LO-

h 2 ′2 − 2 = hω phonon scattering event in the conduction band (k k 2/) me LO , from which

ω = ω′ − ω = + ω the oscillation periods can be found to be is osc 1( me mh ) LO .

During and immediately after an ultra-fast laser excitation, the carriers scatter

mainly due to an unscreened potential [59,60]. The ultra-fast dynamics of semiconductor

in such conditions require considering the quantum kinetic theory with bare Coulomb

51 interactions. The characteristic timescale for the build-up of screening is of the order of the period of Plasmon oscillations [60]. The quantum-kinetic random-phase- approximation (RPA) theory enables to explain quantitatively the density dependence of the decay of FWM signals in the high density regime [61, 62]. At high densities the

-1 1/3 exciton decay time τ in the quantum kinetic regime follows the rule τ ∝ n eh (with n eh being the created electron-hole pair density) [62]. This is well established for bulk materials but has also been found in QWs [62]. However, the screening efficiency might depend on the dimensionality of the system and this leads to the exciton decay time τ as

-1 1/D τ ∝ n eh (with D being the dimensionality of the system) [63].

52 4. Description of the experimental setup

4.1. Sub-30 fs laser; Theory and operation

4.1.1. Laser pulses

A light pulse can be constructed mathematically by multiplying a plane wave with an envelope function. So a Gaussian pulse can be written as

( ) = −( τ )2 ⋅ [ ω ] E t E0 exp[ t G ] exp i 0t 40( )

The Full Width at Half Maximum (FWHM) ∆t of intensity profile I(t) = E(t) 2 of the

∆ = τ ⋅ Gaussian pulse is given by t G ln2 2 . The spectral content of the Gaussian pulse can be obtained through the Fourier transformation which is also a Gaussian function, indicating the larger frequency content comparing to the single unique frequency for a plane wave. So

 τ 2 ω 2  ()ω = τ π − G  ()()ω = ω 2 E E0 G exp  , I E 41( )  2 

Since the temporal and the spectral characteristics of the field are related to each other by

Fourier transformation the bandwidth ∆ω at FWHM of I(ω) and the pulse duration

∆t can not vary independently. As a consequence of the uncertainty principle there is a

minimum bandwidth product and it is given by

2 ⋅ ln2 2 ∆ω ∆t ≥ ×τ ⋅ ln2 2 = ln4 2 42( ) τ G G

∆ν ∆t ≥ .0 4413

The equality holds for pulse without frequency modulation (not chirped), and the pulse is called Fourier-transform-limited pulse [64]. Such pulses exhibit the shortest possible

53 duration at a given spectral width and pulse shape. The table below shows the standard pulse profiles and their uncertainty product value [66].

Field ∆t ∆ω Intensity Profile Spectral Profile ∆ν ∆t Envelope (FWHM) (FWHM) exp[ − (2 t τ )2 ] 1.777 τ exp[ −(τ ω)2 ]2 .2 335 τ 0.441 Gauss G G G G

sec h 2 (t τ ) .1763 τ sec h 2 (πωτ )2 .1122 τ 0.315 sech s s s s

For ultra-short pulses, it is required to generate transform limited pulses to minimize the pulse duration for a given spectral bandwidth. But when short pulse travels through a dispersive medium, the component frequencies are separated in time that results the group velocity dispersion.

4.1.2. Group velocity dispersion (GVD) and its compensation

In a linear medium, frequency components of a propagating pulse travel with different phase velocities that are given by the dispersion relation c(ω) = ω k(ω) . After traveling a certain distance each wave component gains a phase increment ∆φ that induces a phase

modulation in the time evolution of the electric field if the wave number k(ω ) is

ω frequency dependent. Considering the pulse spectrum around the center frequency 0 ,

the spectral phase φ(ω ) can be expanded by Taylor expansion [64, 66]

dφ 1 d 2φ 1 d 3φ φ(ω) = φ(ω ) + (ω − ω ) + (ω − ω )2 + (ω − ω )3 L 0 ω 0 ω 2 0 ω 3 0 d ω 2 d ω 6 d ω 0 0 0

1 1 = φ(ω ) + φ′(ω )( ω − ω ) + φ ′′(ω )( ω − ω )2 + φ ′′′ (ω )( ω − ω )3 L 43( ) 0 0 0 2 0 0 6 0 0

54 φ′ ω φ′′ ω φ′′′ ω Here ( 0 ) , ( 0 ) and ( 0 ) are the derivatives of the phase with respect to

frequency and are known, respectively as the group delay, second-order dispersion (or

group velocity dispersion (GVD)), third-order dispersion (TOD), fourth-order dispersion

(FOD) etc. [65, 64]. The quadratic frequency dependence, resulting in chirp brings

additional frequency components. Since the spectrum of the pulse remains constant, the

spectral components responsible for chirp appear at the expense of the envelope shape

making the pulse broader. The derivatives of the propagation constant used most often in

pulse propagation problems expressed in terms of n are: [66]

d k 1  d n  = n − λ  dω c  dλ  2  λ   2  d k =   1 λ2 d n  2    2  44( ) dω  2π c  c  dλ  3  λ  2  2 3  d k = −  1  λ2 d n + λ3 d n  3   3 2 3  dω  2π c  c  dλ dλ 

Here GDD is the group delay dispersion and TOD corresponds to third-order dispersion.

λ The GVD due to material of length lm , at the central wavelength 0 , can be obtained from the above equation as [66]

λ3l d 2n 0 m 45( ) 2π c2 dλ2

To produce a short pulse it is very essential to control the chirp in pulse and this can be done through a number of designs using prisms and grating pairs [67,68].

In our home built Ti: Sapphire sub-20 fs laser the dispersions are compensated by introducing a prism pair into the system (see fig. 12). In such case the Brewster angle cut fused silica prisms P1 and P2 are placed with the Brewster angle to have minimum loss in

55 the cavity. The exit plane of prism P1 stays parallel to the entrance plane of prism P2 and it hits the high reflector. Due to different phases and group velocities in the prisms, the red spectral components follow longer paths than the blue spectral components, thus the red part is delayed comparing to the blue part providing a negative dispersion. So the prism pair can be used for compression of positively chirped pulses, so the net GVD becomes zero. l Red d Blue Red HR Blue Blue and Red P1 P2

Figure 12: Schematic diagram for GVD compensation using two prisms, after [69].

In the experiment the magnitude of the GVD compensation can be achieved by

changing the distance between the two prisms ( l ) and also by shifting the prism P2

perpendicular to the base line ( d ) [69].

4.1.3. Principle of mode-locking

In a continuous wave multi-mode laser, the modes oscillate independently of each other and have random phases among them. Using the technique of mode-locking, it is possible to have a well defined and constant phase among the different modes. Constructive interference can lead to the formation of wave packet and hence to the formation of laser pulses. The methods of mode locking can be put into three categories: active mode locking, passive mode locking and self mode locking [59, 63].

56 In my project for the production of a sub-20 fs laser pulse the self mode locking

has been used. In the self mode locking process, modes may lock without any need of

external modulation (active mode locking) or using a saturable absorbing medium in the

cavity (passive mode locking). The main principle of self mode locking relies on the Kerr

lens effect, a third order nonlinear effect. In self mode locking the pulse does not go to

the pulse regime spontaneously. To start the pulse regime a rapidly rotating optical slide

can inserted. A quick push to one of the mirrors or a quick movement of the prisms can

bring the laser into the pulsed regime.

4.1.4. Sub-20 fs Ti: Sapphire laser design

The design of our sub-20 fs Ti: Sapphire laser, tunable in the spectral range from

830 – 890 nm, is based on the description of Mode locked-Locked Ti: Sapphire laser,

Henry C. Kapteyn, and Margaret M. Murnane . The design of the cavity follows the standard four-mirror Z-folded resonator for a Kerr-lens mode-locked laser [70]. The cavity contains a Brewster-cut 2 mm thick Ti: Sapphire crystal, two curved mirrors and it has two arms with two plane broadband high reflective end-mirrors. One of the arms with the mirror of reflectivity of 100 % contains two Brewster-cut fused-silica prisms used for the GVD compensation in side the cavity. The other arm has the output coupler which reflects 90% and transmits 10% of the in-cavity laser power (see figure 13). The amplifying medium used in our laser system is the Titanium-doped aluminum oxide

3+ (Ti : Al 2O3). The crystal has hexagonal structure with aluminum in the center and

oxygen atoms in the vertices. In the doping process the Ti 3+ ion substitute for the Al 3+ ion

in the sapphire crystal [64]. It forms a four-level laser medium. The absorption band is in

57 the blue green part of the visible spectrum. The emission band is shifted towards the lower energies and it covers from 650 nm to 1100 nm. Because of the extra large gain

bandwidth of Ti: Sapphire crystals, laser pulse of duration 6 fs can be possible [71].

In our system (see fig. 13) the pumping source is the Spectra-Physics Millennia

Xs continuous wave (cw) 532 nm green laser. It provides the laser in TEM 00 mode and an output power from 0.2 W up to 10 W. Two solid state fiber coupled GaAs diode laser are used to end-pump the neodymium yttrium vanadate (Nd:YVO 4) crystal which provides laser light at 1064 nm wavelength. A non-critically phase-matched lithium triborate

(LBO) placed in the cavity converts the intra-cavity light to green 532 nm wavelength.

The polarization of the green laser has been made parallel to the optics table by using two mirrors in the up and down position (M) that changes the polarization by π 2 . It helps to

have lossless propagation through the Brewster surfaces of the Ti: Sapphire crystal and

the prisms. Using a lens (L) of 15 cm, the green laser is focused into the sapphire rod

collinearly with the laser axis through the back of one of the curved mirrors (M1). To

avoid the accumulation of heat in the sapphire crystal, the supporting mount is cooled by

the running water at a temperature of 19 0 C using a water chiller. The mount is placed on

a rotating stand which helps to find the Brewster angle position. This is obtained by

looking at the reflection of the green light coming from the sapphire crystal and finding

the position of the sapphire crystal for which the reflection is minimum. The concave

dielectric mirrors M1 and M2 are transparent for pumping wavelength 532 nm and their

high reflectivity band ranges from the wavelength of 830 nm to 960 nm. The radius of

curvature of these mirrors is 10 cm.

58 532 nm RM Tsunami Millennia X

HR M2 P2 L M M1 P1

P4 M7 M3 M4 P3 M6 M5

Figure 13: Schematic diagram of the sub-20 fs laser.

M1 has antireflection coating on the back side for pump light. The mirror M1 and M2 are placed in holder which can be translated using the micrometer screws. That helps to find the stability region of the laser [64, 70]. The group velocity dispersion in the cavity is compensated by two pair of Brewster cut fused silica prisms (P1) and (P2). The Brewster angle position of the prisms is determined by rotating the prisms and finding the position

59 for which the reflection of the fluorescence light of the Ti: sapphire coming from the prism surface is minimum. The tip to tip distance between the prisms (P1) and (P2) is 51 cm. The prisms are mounted on a translation stage which enables to tune the center wavelength of the laser and also to shorten the pulse width. The GVD of the output beam due to the output coupler is again compensated by using two extra-cavity prisms P3 and

P4. The tip to tip distance between prisms (P3) and (P4) is 50 cm. The output beam hits the lower part of the curved gold mirror M5 which has a radius of curvature of 140 cm.

This focuses the beam on the high reflector M7 going through prisms P3 and P4. The reflected beam through prisms P3 and P4 hits middle part of gold mirror M5. Then it is picked up by the mirror M6 that sends the output beam to BBO. The arrangement for the

BBO is shown in the fig. 13 which involves a curved gold mirror of radius of curvature 5 mm, a rotation stage that holds the BBO and another curved mirror of radius of curvature of 10 cm to pick up the blue light.

The sub-20 fs laser produces a power more than 1 W for cw-radiation at the

wavelength of 860 nm when it is pumped with 6.5 W and when the cavity is aligned for

the optimal cw-regime. The power is usually measured just after the output coupler. The

fluorescence light from the Ti: Sapphire crystal is put to the edge of the prisms P1 and

P2. By moving the mirror M2 the regime for mode-locking is attainable. Therefore for

several positions of mirror M2 the pulsed regime is seen. Moving the prism P1 very fast

provides a jolt to the system; the cw-laser can be brought to pulse in this regime. By

moving prisms P1 and P2 a very broad pulse can be obtained. The pulsed laser provides

around 550-600 mW power when operated at wavelength 860 nm and 500-550 mW for

wavelength 880 nm. Using the spectrum analyzer the spectrum of the pulse can be seen

60 on an oscilloscope. The full width at half maximum of the spectrum obtained on the oscilloscope is ~45-50 nm depending on the position of the centre wavelength. The center wavelength of the pulsed can be tuned from 840 to 890 nm. The total length of the cavity is 1.55 m and laser operates with the repetition rate of ~96.7 MHz. Using the two-photon absorption technique with a GaAs photodiode, the duration of the pulse has been calculated to 19 ± 1 fs. The autocorrelation technique will be described in the next section. The fundamental pulses are frequency doubled by a 200 µm thick BBO crystal.

The emitted average power after second harmonic generation light is larger than 70 mW at 430 nm and is more than 50 nm for 440 nm. The autocorrelation trace was taken with a

SiC diode based on two-photon absorption. The frequency doubled pulses produced are of ~25 fs pulse width. The pulse in our FWM measurements reveal ~35 fs pulses when then reach the sample inside the cryostat because of the pulse broadening caused by the window of the cryostat, the lens used to focus the beams on to the sample, and because of the beam splitters.

4.2. Pulse width measurement: Autocorrelation technique

The fastest detector that can be used to measure the pulse width of a sub-20 fs pulse in time domain is the pulse itself. The technique is known as the autocorrelation. The general n th order fast correlation function with background for n distinct light pulses

= ξ ω + φ having real electric field amplitudes E j (t) (t)cos[ jt j (t )] is given by after [72 ]

∞ {}+ + τ + L + + τ 2n ∫ E(t) E(t 1 ) E(t n−1 ) dt (n) τ τ Lτ = −∞ g B ( 1 , 2 , n) ∞ 46( ) n ∫ E 2n dt −∞

61 The experimental technique of autocorrelation follows the time-space transformation, that

1 fs corresponds to an optical path of 0.3 m which can be measured and calibrated.

De tector

E(t) E(t −τ )

Figure 14: Schematic diagram of an interferometric autocorrelator.

In an optical autocorrelation the intensity of the laser I(t) = E(t) ⋅ E* (t) can be divided using beam splitter and then the divided pulses can be correlated with each other where one part is delayed. Considering the usual set up for this is the Michelson interferometer technique fig. 14, the first order autocorrelation function is given by [64]

∞ ∞ ∞ g )1( (t) = ∫ {}E(t) + E(t −τ ) 2 dt = 2 ∫ I(t) dt + 2 ∫ E(t)E(t −τ ) dt 47( ) −∞ −∞ −∞

But the full knowledge of E(t ) requires the measurement of higher order correlation functions like the second-order function which can be obtained experimentally using the second-harmonic generation (SHG) or two-photon absorption (TPA) techniques. The second order autocorrelation function is given by

∞ 2 g )2( (t) = ∫ {}(E(t) + E(t −τ ) )(E(t) + E(t −τ ) )* dt 48( ) −∞

62 Two-photon absorption autocorrelations

In this method, a semiconductor detector - a photodiode - is excited at wavelength

below the band-gap cut-off using a two-photon absorption process. The current caused in

the photodiode due to the excited electron-hole pair in the semiconductor can be directly

recorded on an oscilloscope as a function of time t . The current in photodiode is directly proportional to the intensity of incident laser pulse. From equation (48), the second order autocorrelation function becomes

∞ )2( = { + −τ + * −τ + * −τ }2 gTPA (t) ∫ I1 (t) I 2 (t ) E1 (t)E2 (t ) E1 (t)E2 (t ) dt −∞

∞  I 2 (t) + I 2 (t −τ ) + 4I (t)I (t −τ ) + 4I (t)E (t)E (t −τ )cos ωt  =  1 2 1 2 1 1 2  dt 49( ) ∫ + −τ −τ ω + −τ ωτ  −∞ 4I 2 (t )E1 (t)E2 (t )cos t 2I1 (t)I 2 (t )cos 2 

The result in the above equation refers to interferometric TPA autocorrelation where oscillations with ωτ and 2ωτ are seen with different amplitudes. In our setup fig 15,

two laser beams are focused onto the photodiode with a very small angle. This indicates

that weak oscillations might be seen in the autocorrelation signal. Considering TPA

autocorrelation without any oscillation, where only a running grating is formed when the

two beams hit the photodiode with an angle, the second-order TPA AC signal becomes

∞ ∞ )2( = {}+ −τ 2 = 2 + 2 −τ + −τ gTPA (t) ∫ I1(t) I2 (t ) dt ∫ I1 (t) I2 (t ) 2I(t)I(t ) dt 50( ) −∞ −∞

Here I1 and I2 are the intensity of the two laser beams and the first two terms act as a

constant background to the autocorrelation signal. From the autocorrelation signal the

pulse width can be determined using the formula.

∆t ∆t f Gaus = P = .0 7071 and f Sech = P = .0 6482 A ∆ A ∆ tAuto tAuto

Vibrating mirror Piezo driver

Pin hole

Fixed mirror

Photodiode Focusing Mirror Voltage Voltage [V]

Oscilloscope Frequency Amplifier 1 2 3 4 5 Generator Time [s]

The experimental setup for the TPA autocorrelation (NT&C Sub 50 AC) is shown in fig.

15. The laser beam goes through two pinholes to confirm the alignment of the beam. The

TPA autocorrelation signal may saturate at very high intensities. In such a case the two pinholes can be reduced to decrease the intensity of the incident light. The incident laser beam falls equally on a beam splitter composed of pair of mirrors. This provides equal intensity to both of beams reflected from two mirrors. One of the mirrors is attached to a piezo-electric translator which can move 0.45 m with an applied voltage of 100 Volt

and the other mirror is fixed with a stand. The voltage applied to the PZT is a saw-tooth

voltage that is provided by a frequency generator and an amplifier that gives offset to the

voltage. Two mirrors stay parallel to each other but they make an angle of ≈ 22 5. 0 with

the horizontal and the gap between them is of the order ≈ 5.0 mm. The optical delay is accomplished in a very small distance at the reflective beam splitter due to the movement of the PZT mirror. The reflected beams from the two mirrors are focused by an off-axis

64 parabolic mirror onto a photodiode, where the two-photon absorption takes place. The photodiode used for blue light is a GaN photo-diode and for infrared fundamental pulse

840-880 nm GaAsP photodiode has been used. The two-photon absorption leads to generation of the signal on an oscilloscope connected to the photodiode. The autocorrelation signal is seen on the oscilloscope whenever the two beams overlap each other on the photodiode spatially as well as temporally. In a saw-tooth voltage, the voltage rises to the maximum value slowly and then drops immediately to its minimum.

In one cycle, the two pulses overlap two times, one while the ramp is increasing slowly and the other overlaps occurs during the very fast drop of the voltage.

If the laser beam moves a distance ∆L with applied saw-tooth voltage having the peak to

peak time distance TSweep , it can be shown that the velocity of the piezo-stage

× ∆ c × ∆t = 2 L = Auto vel Piezo 51( ) TSweep TAuto

α = 0 ∆ = Since the reflecting mirrors are at an angle 67 5. , the distance L LP sin 67 5.

with LP is the distance the piezo moves with applied voltage from its rest position. So from the equation (51)

2L T 1 2L × z T ⇒ ∆t = P × Auto and ∆t = × P × Auto Auto × P × c sin 67 5. TSweep f A c sin 67 5. TSweep

c is the velocity of light and TAuto is the full width at half maximum of the

autocorrelation signal on the oscilloscope the autocorrelation factor is f A .

In our present experiment, the saw-tooth voltage does not fall immediately as shown in fig. 15, rather it drops slowly with certain time and the applied voltage to the

= PZT is 80 V. With this voltage, the piezo moves LP 36.0 m from its rest position. In

65 such a case the value of TSweep is different. The exact shape and voltage of the applied

saw-tooth voltage as well as the measurement for TSweep is shown in fig. 16 together with the autocorrelation signal obtained on the oscilloscope for a 38 fs laser blue pulse.

Tsweep

80 V

58 fs Voltage[V] T Auto autocorrelationsignal

-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0 100 200 300 Time [s] autocorrelation time [fs]

Figure 16: Applied saw-tooth voltage that shows the actual applied voltage and the measurement of TSweep . Autocorrelation trace of the frequency doubled 23 fs fundamental pulse. Using an autocorrelation factor of 0.65 for sech 2 pulses the frequency doubled pulses have a temporal pulse width of 38 fs.

The duration-bandwidth product for sech 2-shaped pulses and for Gaussian-shaped pulses

is 0.315 and 0.44 respectively. Due to sum-frequency generation in the BBO crystal the

blue pulses are Gaussian-shaped with a duration-bandwidth product of ~0.47. This value

is by a factor of 1.1 larger than the theoretical limit of 0.44. The higher value is attributed

to the non-compensated dispersion in the SHG crystal and within the polymer protection

layer on top of the SiC photodiode.

66 4.3. Four-wave mixing experimental setup

Sub-30 fs blue Laser

CM CM 532 nm Tsunami RM Ti: Sapphire Millennia X BBO RM Lens L 1 L 2

BS BS

Pin-holes

Monochromator and Multi-channel analyzer

Beam Splitter (BS) FWM Signal Removable Mirror (RM)

Curved Mirror (CM) Retro -reflectors on translation stages

L 3 L 4 L 5 Cryostat

Figure 17: Schematic diagram of the experimental setup for FWM experiment.

A millennia solid state laser at the wavelength of 532 nm with a power of 7-9 W is used as the pump for the mode locked Ti-sapphire laser that provides pulses of spectral width of 22 meV at a repetition rate of 82 MHz. The wavelength of Ti-sapphire laser can be varied from 700 nm to 1050 nm using different optic sets. The beam from the Ti- sapphire laser is further frequency doubled using a BBO crystal. The temporal width of the frequency doubled pulses was determined to 103±4 fs using the autocorrelation technique (described earlier) that is based on the two-photon absorption in a SiC photodiode. The Ti-sapphire laser pulses with central wavelength 882 nm and of power

1.3 W provides 180 mW of frequency doubled power that has been used to excite the excitons resonantly in a 10 nm ZnSe quantum well. The four-wave mixing experiments,

67 which involve three excitation pulses k1 , k 2 and k 3 , have been performed in the back scattering geometry with the sample mounted in a helium bath cryostat to achieve the required low temperatures. The 1 / e2 focus diameter of the pulses on the sample was

measured 100 µm. The retro-reflectors mounted on translation stages have been used to

create delay between different incident beams. The zero delay between pulses k1 and k 2

as well as between k1 and k 3 has been determined by the maximum contrast of light

interference pattern appearing on the sample surface during temporal overlap of the

pulses. The time integrated and spectrally resolved 2-beam and 3-beam four-wave mixing

− + − signal was detected in direction 2k 2 k1 and k 3 k 2 k 1 respectively. Two

diaphragms are used to block the reflected lights and photoluminescence of beams k1

and k 2 from the sample surface as well as the reflection from the cryostat window.

Finally the signal has been analyzed by a combination of a spectrometer and an optical

multi-channel analyzer. The experiments have been performed for various polarization

configurations by using λ 2 and λ 4 plates in three incident pulses. In the present

document the notation(σ +σ +σ + ) , (σ +σ +σ − ) and (σ +σ −σ + ) indicates the polarization

of the k1 , k 2 and k 3 pulse, respectively. In the performed 2-beam (3-beam) four-wave

τ > τ > mixing experiments a positive delay 12 0 ( 13 )0 refers to the case where pulse k 2

τ < τ < (k 3 ) arrives last whereas a negative delay ( 12 )0 ( 13 )0 refers to pulse k 2 ( k 3 ) arriving first.

68 5. Four-wave mixing experimental results

5.1. Study of interband and interexciton coherences

5.1.1. Three-beam FWM experiments

The first experimental investigation performed is the study of interband and

interexciton coherences in the 10 nm Zn 0.94 Mg 0.06 Se/ZnSe single quantum well using the

polarization depenedent three pulses degenarate FWM experiment.

Xh ( σ+σ+σ+ ) + + - XX ( σ σ σ ) m Xl ( σ+σ-σ+ ) XX m XX h FWMsignal

2.80 2.81 2.82 energy [eV]

+ + + Figure 18: Spectrally resolved 3-beam FWM signals for polarization configurations (σ σ σ ) , + + − + − + (σ σ σ ) and (σ σ σ ) at 55K, taken at a delay time τ = -0.55ps.

+ − Fig. 18 shows the spectrally resolved FWM signal in k 3 k 2 k 1 direction obtained from the 10 nm ZnSe SQW at a delay time τ = -0.55 ps for configurations(σ +σ +σ + ) ,

σ +σ +σ − σ +σ −σ + -2 ( ) and( ) . The intensity of pulses k1 and k 2 together was 1.3 MW cm

-2 and the intensity of the third pulse k 3 was 0.3 MW cm . The center of the 22 meV broad

excitation pulse was set to 2.808 eV, in order to avoid continuum contributions and

predominantly excite the 11 H Xh heavy-hole exciton at 2.806 eV and weakly excite the

69 light-hole exciton 11 L Xl exciton transition at 2.821 eV. In addition the heavy-hole

∆ biexciton induced signal XX h at (2.806 eV - h ) [42] and the mixed heavy-light-hole

∆ ∆ biexciton induced signals XX m at (2.806 eV - m ) and (2.821 eV - m ) [46] are observed,

both being identified from their polarization dependence as described in the next

∆ paragraph. In the parentheses h = 4.8 meV is the binding energy of the heavy-heavy-

∆ hole biexciton, m = 2.8 meV refers to the binding energy of the mixed heavy-light-hole

biexciton. The weak 12H heavy-hole exciton transition visible at 2.815 eV was not considered in the following treatment. Two-pulse four-wave mixing experiments further indicate that the exciton transitions are nearly homogeneously broadened in this ZnSe

SQW [73, 74].

The FWM responses as a function of delay time τ for configurations(σ +σ +σ + ) ,

+ + − + − + (σ σ σ ) and (σ σ σ ) at the energetic position of the Xh transition and for the

+ − + (σ σ σ ) configuration at the biexciton induced XX h transition are displayed in fig 19.

τ < For negative delay time( )0 , pulse k 3 builds up exciton polarizations, which, together

with the subsequent pulses k1 and k 2 leads to the generation of a FWM signal into the

+ − τ direction of k 3 k 2 k 1 after time .

In all configurations the observed FWM signal decays with the dephasing rate of

the interband exciton polarization. While clear quantum beats between heavy- Xh and

light-hole Xl excitons with a period ∆∆∆τ ≈ 270 fs (corresponding to an energy splitting of

∆ = ∆τ ≈ σ +σ +σ + σ +σ −σ + ∆∆Ehl h / ∆∆ 15 meV) are observed in ( ) and ( ) configuration, no

oscillations are visible in (σ +σ +σ − ) configuration. The observed quantum beats reveal

70 constant oscillation amplitude indicating very similar Xh and X l single exciton interband dephasing times.

1 Xh ( σ+σ+σ+) -1 10 ( σ+σ+σ-) ( σ+σ-σ+) -2 σ+σ-σ+ 10 ( ) XX h FWM signal 10 -3

8 6 4 2 0 -2 -4 -6 delay τ [ps]

Figure 19: FWM traces at the spectral position of the heavy-hole Xh exciton transition for different polarization configurations and at the heavy-hole biexciton induced transition XX h for configuration + − + (σ σ σ ) at 55 K.

τ > σ +σ +σ + σ +σ +σ − For positive delay time ( )0 in configurations ( ) and ( ) pulses k1

and k 2 have the same circular polarization so that an exciton population grating with

− grating vector k 2 k1 is generated. The electric field of the delayed pulse k 3 (with same or different circular polarization) generates a nonlinear FWM signal into direction

+ − k 3 k 2 k 1 . The FWM signal persists as long as the grating exits i.e. the signal decays

with the inverse of the exciton lifetime T1 and decreases also due to the diffusion of

excitons within the population grating leading to a reduced contrast of the population

+ + + grating with increasing delay time τ [75]. Only in (σ σ σ ) configuration Xh-Xl quantum beats with damped oscillating amplitude are observed. Furthermore the FWM signal in (σ +σ +σ + ) configuration shows an increase near zero delay, which is not

observed in (σ +σ +σ − ) configuration. As outlined in the next paragraph this behavior is

71 explained by excitation induced dephasing (EID) which is present for (σ +σ +σ + )

+ + − polarized fields but is inactive in (σ σ σ ) configuration.

The FWM signal observed in (σ +σ −σ + ) configuration for τ > 0 behaves

significantly different. In this case pulses k1 and k 2 cannot create a population grating

because they act on different σ + and σ − exciton transitions. Instead they create an interexciton coherence between σ + and σ − excitons that has a structure of an

− orientational grating with momentum k 2 k1 . Again together with k 3 a nonlinear FWM

+ − signal is generated into direction k 3 k 2 k 1 . The nonlinear signal decays with the

same rate as the loss of coherence between two σ + and σ − excitons, thus enabling the determination of the not otherwise accessible interexciton exciton dephasing time. A closer look to the FWM trace, however, shows that the decay cannot be fitted with a single exponential function. Also the Xh-Xl beating clearly observed in this configuration

shows a slight damping for positive delay. In the following paragraph these observations

are explained by the decoherence between Xh and Xl excitons of opposite spin as well as by EID both contributing to the FWM signal at resonance X h. The FWM trace at the biexciton induced XX h transition is (up to third order perturbation) neither affected by the

Xh-Xl interexciton decoherence nor by EID so that the Xh interexciton dephasing time can be directly evaluated from this signal.

The FWM traces at the energetic position of the Xl exciton (see fig. 20) show very

+ − + similar features as observed at the Xh transition energy. In (σ σ σ ) configuration for

τ > 0 the decay rate of FWM signal is faster as compared to the decay rate of the FWM

72 signal at the spectral position of Xh indicating a faster loss of the coherence between two

+ − σ and σ light-hole excitons Xl as well as between Xh and Xl excitons of different spin.

1 ( σ+σ+σ+) ( σ+σ+σ-) 10 -1 ( σ+σ-σ+) Xl

10 -2 FWMsignal 10 -3

8 6 4 2 0 -2 -4 -6 delay τ [ps]

+ − Figure 20: FWM traces into k 3 k 2 k1 direction at the spectral position of the light-hole Xl exciton + + + + + − + − + transition for polarization configurations (σ σ σ ) , (σ σ σ ) and (σ σ σ ) at 55 K.

5.1.2. Theoretical model

Numerical calculations have been performed using equations (37) and (38) for configurations(σ +σ +σ + ) , (σ +σ +σ − ) and (σ +σ −σ + ) . The energetic positions of

exciton and biexciton induced transitions were taken from the experimental FWM

spectra. From the FWM traces we evaluated the combined lifetime Tc to Tc = 35 ps as

γ = γ = -1 γ = γ = -1 well as the interband dephasing rates 21 31 75.0 ps and 61 71 75.0 ps . The

γ = -1 interexciton dephasing rate 32 85.0 ps between two heavy-hole excitons of opposite

+ − + spin was extracted from the (σ σ σ ) trace at the XX h transition. Dephasing rates

γ = γ = -1 63 72 2.1 ps between heavy and light-hole excitons with opposite spin were

+ − + approximated from the (σ σ σ ) FWM trace at the XX l transition. We assumed the

73 same dephasing rates between heavy and light-hole excitons with equal spin e.g.

γ = γ = -1 62 73 2.1 ps .

Xh ( σ+σ+σ+ ) XX m + + - X ( σ σ σ ) XX l m ( σ+σ-σ+ ) XX h FWMsignal

2.80 2.81 2.82 energy [eV]

+ + + + + − Figure 21: Calculated 3-beam FWM spectra for polarization configurations (σ σ σ ) , (σ σ σ ) and + − + (σ σ σ ) for the same conditions as in fig. 18.

γ For simplicity we further assumed that the dephasing rates ts between two-exciton and

γ single exciton states are equal to 21 . The magnitude of the ground to heavy-hole dipole

µ = µ matrix elements 21 31 and that of the ground to light-hole exciton dipole matrix

µ = µ elements 61 71 were set to 1 and 0.57, respectively, according to their relative

µ 2 µ 2 oscillator strength-ratios 21 : 71 of 3:1 in relation to their valence band functions.

Furthermore the spectral shape of the excitation pulse was considered as described in ref.

(29). The exciton-exciton scattering parameter β was determined from intensity

− dependent measurements to β = 22 s 1 cm 2 as outlined in the following paragraph. The

γ µ = ⋅ 10 -2 still unknown parameters N , 67 , and ts were fitted to N 8.2 10 cm ,

74 γ = −1 µ = 76 2.1 ps and ts 3.0 for all two-exciton transitions to give reasonable agreement

between experimental data and calculated spectra and traces.

As demonstrated in fig. 21 the calculated FWM spectra for configurations

(σ +σ +σ + ) , (σ +σ +σ − ) and (σ +σ −σ + ) at a delay time τ = - 0.55 ps reveal the same

polarization dependence as observed in the experimental spectra shown in fig. 19. In

addition the relative signal intensities between the different exciton and biexciton

transitions as well as between different polarization configurations are well reproduced.

1 Xh

σ+σ+σ+ 10 -1 ( ) ( σ+σ+σ-) ( σ+σ-σ+) 10 -2 σ+σ-σ+ ( ) XX h FWM signal 10 -3

8 6 4 2 0 -2 -4 -6 delay τ [ps]

Figure 22: Calculated 3-beam FWM traces at the energetic position of the heavy-hole Xh exciton transition + + + + + − + − + for configurations (σ σ σ ) , (σ σ σ ) and (σ σ σ ) and at the biexciton induced transition XX h for + − + configuration (σ σ σ ) for the same conditions as in fig. 19.

The calculated FWM traces at the heavy-hole exciton transition Xh and at XX h are shown in fig. 22. According to the theoretical expression for (σ +σ +σ + ) configuration at τ > 0

the slow signal decay at long delay times is determined by the combined lifetime Tc .

Occurring oscillations close to zero delay are generated by a term containing an exponent

Ω* τ exp( iΩΩ62 ) and (weaker) by EID terms containing Xh and Xl transitions. The complex

75 Ω* = ω + γ σ + σ + frequency ΩΩ62 62 i 62 indicates the coherence between -Xh and -Xl excitons

ω γ with a beating frequency of 62 . Dephasing rate 62 is responsible for the observed

damping of the beats. The rise of the FWM intensity near zero delay is caused by EID

since pulses k1 and k 3 create an exciton grating so that the k 2 exciton polarization is

+ − τ < ω diffracted into direction k 3 k 2 k 1 . The contributing terms for 0 at resonance 21

ΩΩΩ τ ΩΩΩ τ contain the exponential functions exp( i 21 ) and exp( i 61 ) , leading to the

γ γ appearance of Xh-Xl quantum beats. Since dephasing rates 21 and 61 were chosen

equal the FWM trace shows a single exponential decay and a constant oscillation

amplitude with delay τ in agreement to the experimental observation.

+ + − In (σ σ σ ) configuration for delay τ > 0 the calculated FWM polarization is

−τ again governed by the exponential function exp( / T1) . There is no term accounting for

ω quantum beats at resonance 31 . Since the polarization directions are opposite for pulses

k1 and k 3 EID is inactive for pulse k 2 . Hence no EID signal close to zero delay is

generated in agreement to the experimental observation. In contrast local field effects

σ +σ +σ − (LFE) arising from pulses k 2 and k1 are active in ( ) configuration and can lead to an additional FWM signal close to delay zero [74]. Since we do not observe a noticeable signal increase in our experiment we conclude that local field effects are of minor importance in consistence to our previous investigation [73]. From the experimental polarization dependence we conclude that LFE are of little importance as already observed in previous investigations [73]. The contributing terms for τ < 0 at

76 τ resonance Xh contain only an exponential function exp( i 21 ) , explaining the lack of Xh-

Xl quantum beats in this polarization configuration.

1 ( σ+σ+σ+) ( σ+σ+σ-) -1 10 + - + ( σ σ σ ) Xl

10 -2

FWMsignal 10 -3

8 6 4 2 0 -2 -4 -6 delay τ [ps]

Figure 23: Calculated 3-beam FWM traces at the energetic position of the light-hole Xl exciton transition + + + + + − + − + for configurations (σ σ σ ) , (σ σ σ ) and (σ σ σ ) for the same conditions as in fig. 20.

In (σ +σ −σ + ) polarization configuration for τ > 0 the calculated FWM polarization at

ω − ΩΩΩ τ resonance 31 is determined by terms containing exponents exp( i 32 ) and

Ω* τ γ exp( iΩΩ63 ) as well as EID (with dephasing rate 2 21 ), thus leading to a multi

exponential decay of the FWM signal and a damping of the Xh-Xl oscillation amplitude.

In contrast the FWM polarization at the XX h is generated by a term containing only the

− ΩΩΩ τ τ < exponent exp( i 32 ) . As mentioned earlier EID is inactive at transition XX h. For 0

ΩΩΩ τ the leading terms at resonance Xh contain the exponential functions exp( i 21 ) and

ΩΩΩ τ exp( i 61 ) , explaining the appearance of Xh-Xl quantum beats. The calculated FWM

traces at the energetic position of the light-hole exciton transition Xl are demonstrated in

+ + − fig. 23. The traces for (σ +σ +σ + ) , (σ σ σ ) configuration are very similar to the

77 traces calculated at the Xh resonance and can be explained in a similar way. In polarization configuration (σ +σ −σ + ) for τ > 0 the calculated FWM polarization at

− ΩΩΩ τ resonance Xl is determined by terms containing exponential functions exp( i 72 ) ,

− ΩΩΩ τ exp( i 76 ) and by EID terms. Due to the larger dipole moments, the magnitude of the

− ΩΩΩ τ − ΩΩΩ τ exp( i 72 ) term is significantly larger than that of the exp( i 76 ) term so that an

γ exact determination of the interexciton rate 76 from the experimental data is very

difficult. However, the experimental trace can be modeled consistently assuming

γ = γ 76 72 .

5.1.3. Intensity dependent measurements

Despite some deviations from experimental results as e.g. the weaker FWM signal

+ + − + + + in (σ σ σ ) configuration for τ > 0 or the stronger Xh-Xl beating in (σ σ σ )

configuration for τ < 0 , the model calculations successfully reproduce the experimental data and significantly contribute to the understanding of the complex FWM processes. In comparison with theory the experiments also provide reliable values for the interband

γ γ + dephasing rate 21 (N3 ) as well as for the interexciton dephasing rates 32 (N1 N2 )

γ + and 72 (N1 N2 ) . However, since these values are obtained at different exciton

densities a direct comparison between these dephasing rates is not admissible.

We therefore performed intensity dependent measurements in (σ +σ −σ + )

configuration using the same experimental setup. In the first measurement series the

-2 intensity of pulses k1 and k 2 was kept constant at 1.3 MW cm and the intensity of the

78 third pulse k 3 was varied. In the second series the intensity of pulse k 3 was kept

-2 constant at 0.3 MW cm and the intensities of pulses k1 and k 2 were varied. These

γ 0 γ 0 γ 0 measurements enable us to determine the dephasing rates 21 , 32 , 72 at zero exciton

density and provide values for the exciton-exciton scattering parameters

β β β 21 , 32 and 72 .

σ+ σ- σ+ X ( ) h (a) (b)

I3/I 03 2.0 Xh

XX h 30 Xl 1.5 ]

15 -1 [ps

7 21 γ 1.0 4

2 normalized FWMnormalized signal 1 0 1 2 3 4 4 2 0 -2 -4 density N [10 10 cm -2 ] delay τ [ps] 3

+ − + Figure 24: (a) Normalized 3-beam FWM traces for (σ σ σ ) configuration at the X exciton transition r h for a fixed k1 and k 2 pulse intensity and variable k 3 pulse intensity labeled as ratio I 3/I 03 . (b) Measured

dephasing rates from FWM traces at the Xh, XX h and Xl transition energy as a function of the exciton γ density N3 generated by pulse k3 The dashed line gives a linear increase of the dephasing rate 21 with an β = −1 2 exciton-exciton scattering parameter of 21 22 s cm .

Fig. 24(a) shows the normalized FWM traces for (σ +σ −σ + ) configuration at

position Xh. The varied intensity of the third pulse k 3 is specified as ratio I 3/I03 with

-2 τ < respect to a reference intensity I 03 = 0.6 MW cm . For 0 pulse k 3 arrives first so that exciton density N3 is responsible for the density dependent dephasing of the nonlinear

γ = γ 0 + β τ > γ + response according to 21 (N3) 21 21 N3 . For 0 the decay rate 32 (N1 N2 )

79 of the FWM polarization remains nearly constant since the exciton density created by

pulses k1 and k 2 is constant. The observed increase of the FWM signal close to zero

delay is attributed to higher order FWM processes which are not further discussed in this

context.

γ Fig. 24(b) depicts the extracted dephasing rates 21 (N3) as a function of the

exciton density N3 . The two-dimensional exciton density created by an excitation pulse with intensity 1 MW/cm 2 was estimated to 5.2 ⋅10 9 cm -2 [73]. Up to an exciton density

= ⋅ 10 -2 γ of N3 2 10 cm we find a linear dependence of dephasing rate 21 (N3) in

β ≈ −1 2 accordance to an exciton-exciton scattering parameter 21 22 s cm . Above

= ⋅ 10 -2 N3 2 10 cm the density dependence begins to saturate due to bleaching which is

in agreement to FWM experiments performed on GaAs quantum wells at high excitation

γ 0 γ 0 = −1 intensities [76]. Finally, a dephasing rate 21 was extrapolated to 21 73.0 ps . Fig.

24(b) also includes the dephasing rates obtained from traces at position XX h and Xl. They reveal similar density dependence in particular nearly equal zero-density dephasing rates

γ 0 ≈ γ 0 β ≈ β 21 61 as well as equivalent exciton-exciton scattering parameters 21 61 for Xh

β ≈ β and Xl exciton transitions. The same scattering parameters 21 61 and zero density

γ 0 ≈ γ 0 rates 21 61 are found if the constant k1 and k 2 intensities are changed in these

experiments.

Fig. 25(a) shows normalized FWM traces for configuration (σ +σ −σ + ) varying

-2 the intensity of k1 and k 2 with I 01 +I 02 = 0.7 MWcm being the reference intensity for

80 the different traces. In this situation the dephasing rate of the FWM signal remains constant for τ < 0 (constant exciton density N3) while the dephasing rate for τ > 0

increases with increasing intensities of pulses k1 and k 2 . Deviations from this behavior

are again attributed to χ )5( and higher order effects.

σ+ σ- σ+ ( ) Xh (a) (b)

(I 1+I 2)/(I 01 +I 02 )

34 Xh

4.0 XX h 16 Xl ] 9 -1 3.0 [ps 72

5 γ , ,

32 2.0 2 γ

1 1.0 normalized FWMsignal

4 2 0 -2 -4 0 2 4 6 10 -2 delay τ [ps] density N1+N2 [10 cm ]

+ − + Figure 25: (a) Normalized 3-beam FWM traces for (σ σ σ ) configuration at the Xh exciton transition

for a fixed k3 pulse intensity and variable k1 and k 2 pulse intensity labeled as ratio (I 1+I 2)/ (I 01 +I 02 ). (b)

Measured dephasing rates from FWM traces at the Xh, XX h and Xl transition energy as a function of the exciton density N1+N 2 generated by pulses k1 and k 2 . The dashed and dotted-dashed lines give a linear γ γ β = −1 2 increase of the dephasing rate 32 and 72 with scattering parameter 32 22 s cm β = −1 2 and 72 60 s cm , respectively.

Fig. 25(b) depicts the extracted dephasing times at spectral positions Xh and XX h. At position Xh the extracted values represent an average dephasing rate involving the

γ γ γ dephasing rates 32 (dominating term), 63 and 2 21 (from EID). The rate at position

γ XX h only accounts for the interexciton dephasing rate 32 and is therefore somewhat lower as compared to the average value obtained from the Xh trace. In linear

γ approximation the density dependence of dephasing rate 32 is given by

81 γ + = γ 0 + β + + ≈ ⋅ 10 -2 32 (N1 N2 ) 32 32 (N1 N2 ) . Up to a density of N1 N2 2 10 cm we

γ find a linear increase of 32 in accordance to an exciton-exciton scattering parameter

β ≈ −1 2 32 22 s cm indicating the same Coulomb scattering efficiency for heavy-hole

+ = ⋅ 10 -2 interband and interexciton coherences. Above density N1 N2 2 10 cm the

dependence starts to saturate as already observed in the measurements before. The zero-

γ 0 γ 0 = −1 density dephasing rate 32 was determined to 32 8.0 ps indicating similar exciton-

phonon and exciton-impurity scattering mechanisms for both types of coherences. The

β γ 0 same scattering parameter 32 and zero density rate 32 are found if k 3 intensity is changed in these experiments.

Fig. 25(b) also contains dephasing rates which have been extracted from the

FWM trace at the spectral position of Xl. From similar arguments as at position Xh the

γ γ extracted rates are average values between rates 72 and 2 21 . The contribution of the

γ 76 containing term can be neglected due to its weak dipole matrix elements. In

comparison with our model calculations we estimate an exciton-exciton scattering

β ≥ −1 2 parameter of 72 60 s cm . This significantly stronger exciton-exciton interaction

for the Xh-Xl interexciton coherence as compared to the Xh-Xh coherence is also noticeable in fig. 25(a) by the increasing damping of the Xh-Xl oscillations with rising exciton

γ 0 γ 0 = −1 density. Finally, the zero-density rate 72 is estimated to 72 9.0 ps indicating

similar exciton-phonon or exciton-impurity interactions for all coherences. The enhanced

β exciton-exciton scattering of rate 72 might be explained by an increased Pauli repulsion

82 + − + − + between σ Xh - σ Xl coherences and σ Xh excitons (or between σ Xh - σ Xl

− coherences and σ Xh excitons), where all interacting excitons carry electrons of equal

= − = + spin momentum ms 2/1 (or ms 2/1 , respectively).

γ Although an evaluation of the interexciton dephasing 76 is not feasible from our

γ 0 γ 0 experiment we may expect that the zero-density dephasing rate 76 is similar to 32 .

Since only two electrons of equal spin momentum are involved in Xl-Xl coherence

scattering events (same as for the Xh-Xh interexciton coherence) we might further

β conclude that the exciton-exciton scattering parameter 76 is smaller than that of the Xh-

Xl coherence. Further spectrally resolved three-pulse FWM experiments using spectrally

narrow 1 ps pulses for individual excitation of either Xh or Xl exciton transitions may help to clarify these assumption.

83 5.2. Carrier correlated dephasing of excitons in 10 nm ZnSe QW

5.2.1. Two-beam experiments

The FWM spectrum obtained at delay τ ≈ 0 in ( ↑↑) configuration is shown in

Fig. 26. In addition the reflection spectrum of the incident laser pulse is given as dashed

2 line. The intensities of pulses k1 and k 2 were 2.2 and 3.0 MW/cm , respectively. The

center of the pulse energy was set to 2.805 eV. Due to the large spectral width of the

excitation pulse various 1s QW exciton states are clearly visible. Most prominent are the

heavy-hole exciton transitions 11h at 2.806 eV, 12h at 2.815 eV, 13h at 2.827 eV as well as the light-hole exciton transition 11l at 2.821 eV. The transitions were identified by

comparison with calculated exciton energies using the envelope function approximation

[7] where heavy and light-hole excitons are considered to be decoupled.

11h pulse reflectionpulse 11l

X 13h h FWM signal XX 1 23h k

- 12h 2 12l k 22l 2 22h

2.80 2.82 2.84 2.86 2.88 energy [eV]

τ ≈ Figure 26: Two-beam FWM spectrum (full line) of a 10 nm thick ZnSe SQW at delay 12 0 at 55 K using 30 fs excitation pulses in ( ↑↑) configuration. Excited exciton transitions are labeled. The dashed line shows the reflection spectrum of the incident laser pulse. The peak energy was centered at 2.805 eV. The 2 intensities of pulses k1 and k 2 were 2.2 and 3.0 MW/cm , respectively.

The calculations were performed using a strain free and dimensionless conduction band parameter Qc = 0.3. A description of the performed calculations as well as used material

parameters for zincblend ZnSe and MgSe are given in [77]. Higher excitonic transitions

84 that energetically lie within the 11h and 11l exciton continua are more difficult to

determine due to the mixing of bound and continuum states. The weak transition at 2.84

eV is most likely assigned to a combination of a 12l and 22h transition according to their energetic position using above mentioned calculations. The stronger transition at 2.854 eV is assigned to a 23h exciton transition. The subsequent transition at 2.857 eV is

attributed to the heavy-hole exciton state Xh of the Zn 0.94 Mg 0.06 Se barrier [78] as

evidenced by photoluminescence excitation (PLE) measurements (not shown here). The

enhancement of the 23h transition is attributed to a resonant Xh-23h state mixing. The

weak transition at 2.869 eV is ascribed to a 22l transition. The light-hole transition Xl of

the Zn 0.94 Mg0.06 Se barrier becomes visible in the laser reflection spectrum and is also

observed in PLE measurements at 2.880 eV but it is too weak to be detected in the FWM

spectrum. Furthermore transition XX is assigned to the 11h exciton-to-biexciton transition

by comparison with earlier FWM experiments on a similar structure [73].

13h 11l 22h 23h

FFT intensity FFT 0 20 40 60 energy [meV] FWM signal 1 k - 2 k 2

2 1 0 -1 delay τ [ps] 12

85 Figure 27 shows the FWM trace at the energetic position of the 11h exciton transition on a logarithmic scale. The 11h exciton decay reveals a fast and non-exponential decay for

τ < pulse delays shorter than 12 500 fs. Within this time interval it is impossible to relate

the decay rate to the spectral FWHM of the 11h exciton transition (~ 0.8 meV) via

Fourier transformation. The prompt decay of the signal near pulse overlap is therefore

attributed to the destructive interference between 11h transitions and 11h continuum states [79,80] (mainly continuum edge states where the spectral pulse intensity is still high). Correspondingly the first 11h -23h beating maximum that should appear at

τ = τ ≈ 12 85 fs is strongly suppressed. After 12 190 fs the 11h transition and eh -pairs at

the continuum edge are in phase again leading to a pronounced 11h -23h beating

τ = τ > maximum at 12 170 fs. For delays 12 300 fs the contribution of coherent eh -pairs to the FWM signal significantly decreases due to fast phase destroying scattering and relaxation processes within the continuum. Also the 11h -23h beating is strongly damped

due to scattering of 23h states with continuum states and due to relaxation into lower

τ > lying exciton states. For longer delays 12 500 fs the FWM trace shows a pronounced

11h -11l beating and the decay becomes almost single exponential. As expected from

γ = −1 Boltzmann kinetics the decay rate 2 5.0 ps correlates to the observed FWHM of the

Γ ≈ Γ = γ exciton transition of 7.0 meV via h 2 2/ .

The inset of figure 27 shows the Fourier transform (FT) spectrum of the 11h

FWM trace. It clearly reproduces transitions 11l , 13h and also reveals the weak

transitions 12l , 22h as well as a broad peak separated ~50 meV from the 11h transition

that is attributed to the resonantly enhanced and rapidly decaying 23h exciton transition.

86 Furthermore the weak signature around 7 meV is attributed to the 12h transition. The FT spectrum does not exhibit oscillations of energy 31.6 meV as reported in [81]that are described by LO-phonon-exciton quantum kinetics. However, these oscillations might be obscured by the broad 12l , 22h peak occurring at ~30 meV. Oscillations with 2LO

phonon frequency [82] are not expected under present excitation conditions.

τ < A quantitative modeling of the FWM signal within time interval 12 500 fs with probably non-Markovian decay requires quantum kinetic calculations including correlated carrier-carrier as well as correlated two-pair scattering processes [83,84,85] but is beyond the scope of this investigations. Approximately we performed calculations that are based on the optical Bloch equations for a homogeneously broadened five-level system up to third order for δ-shaped laser pulses. In these calculations we neglect phenomenological extensions like excitation induced dephasing and biexciton formation.

− After Fourier transformation the third-order polarization into direction 2k 2 k1 reads

2 2  6 Ω* τ  )3(  2M i1 (M j1 exp( i j1 ))  P (ω,τ > )0 ∝exp( iωτ ) ⋅ ∑ (52) FWM  ω − Ω  i, j>1 i1 

Ω = ω − γ with resonance energies i1 i1 i i1. Four of the considered levels i are the initially occupied ground state and excitons 11h, 11l, 13h and 23h . The fifth level introduces a

rapidly dephasing “ eh -pair state” at the 11h continuum edge (~20 meV above the 11h

ω transition energy). Transition energies i1 were determined from the FWM spectrum

(see fig. 26); the dephasing rates of transitions 11h and 11l were determined to

γ = γ = −1 21 31 5.0 ps from the FWM trace. For transitions 13h and 23h dephasing rates

γ = −1 γ = −1 41 0.1 ps and 51 5.2 ps were used. The decay rate of the artificial “ eh -pair

87 γ = −1 = 2 state” was chosen to 61 0.5 ps . For levels i 2K 4 the oscillator strengths M i1

were adjusted to fit the signal height in the FWM spectrum resulting in ratios

2 2 2 = M 21 : M 31 : M 41 5.0:2.1:3 . The oscillator strengths of the 23h transition as well as

2 = 2 2 = 2 of the “ eh -pair state” were chosen to M 51 3.1 M 21 and to M 61 7.2 M 21 to fit the

FWM trace. The calculated trace is shown in fig 27 as thin solid line. For better visibility

the calculated trace is slightly offset to the experimental curve. Despite the high oscillator

strengths of the 23h transition and the “ eh -pair state” it is not possible to completely fit

τ < the trace within interval 12 500 fs. The too high oscillator strength and the yet strong

deviation at pulse overlap are attributed to quantum kinetic effects that are not considered

in our model. Nevertheless, our limited model is able to provide the essential features of

the FWM trace.

11h

k k A 1 2 FWMsignal 1 k - 2 k 2 2.0 1.5 1.0 0.5 0.0 -0.5 delay τ [ps] 12

Figure 28: FWM traces at the spectral position of the 11h exciton transition for collinear ( ↑↑) and cross- ↑→ 2 linear ( ) polarized fields at 55 K. The intensity of pulses k1 and k 2 were 2.2 and 3.0 MW/cm , respectively.

To gain more information on the quantum coherent processes involved we compared

FWM signals that were obtained in ( ↑↑) and ( ↑→) configuration. The traces at the 11h

88 exciton transition are depicted in Figure 28. The most striking difference occurs at τ ≈ 0

where the strong peak observed in ( ↑↑) configuration significantly decreases (by a factor

of ~50) in ( ↑→) configuration. This strong reduction of the FWM signal signalizes the

spin dependence of quantum kinetic processes involved and suggests a transient

photorefractive effect due to the generation of an e-h pair density grating. This

contribution vanishes for cross-linear fields ( ↑→) while polarization independent

processes like Pauli blocking are responsible for the remaining signal. It is worth noting

↑→ τ < that the signal strength in ( ) configuration is stronger for negative ( 12 0) than for

τ > positive delay ( 12 0 ) which is in contrast to previous investigations where only the

11h exciton state was excited. This observed enhancement is tentatively attributed to the contribution of mixed two-pair correlations (mixed biexcitons ) between 11h excitons and

higher excitons ( 11l, 23h …). The observed signal reduction in the Boltzmann regime

going from ( ↑↑) to ( ↑→) polarized fields is attributed to excitation induced dephasing

(EID) caused by scattering at the created exciton density grating. As expected the

quantum interference between the 11h and 11l transition shows a shift by π going from

(↑↑) to ( ↑→) configuration (indicated by dashed vertical lines) confirming the heavy-

and light-hole character of the assigned transitions.

To study the exciton-eh -pair scattering in the QW we performed intensity

dependent FWM measurements. The measurements were performed in co-linear ( ↑↑)

-2 configuration with equal k1 and k 2 pulse intensities ranging from 0.5 to 8.1 MWcm .

Figure 29 shows intensity dependent FWM traces at different k1 pulse intensities as

89 labelled. In the non-exponential regime a determination of the decay rate is difficult due to pronounced 11h-23h quantum beats.

2.0 ] -1 1.0 [ps

2 pulse intensity γ -2 0 1 2 [MWcm ] n [x10 10 cm -2 ] eh 0.5

1.3

3.6 FWM FWM signal 8.1 2 1 0 delay τ [ps] 12

Figure 29: Normalized FWM traces at the spectral position of the 11h exciton transition for polarization ↑↑ -2 configuration ( ) at 55 K with k1 pulse intensities ranging from 0.5 to 8.1 MWcm . The inset shows

extracted dephasing rates as a function of the eh-pair density created by pulse k1 . The dashed line gives a γ linear increase of the dephasing rate 2 according to eq. (53) using an exciton-eh-pair scattering parameter β = −1 2 of X −eh 90 s cm .

Qualitatively we find that with increasing pulse intensity the contribution of continuum states becomes less important at delays beyond pulse overlap which is attributed to an increasing dephasing of eh -pairs. Also the 23h transition is increasingly decaying as

shown by the stronger damping of 11h-23h quantum beats. In the Boltzmann regime the

γ exciton dephasing rate 2 increases with increasing excitation intensity according to

γ = γ + β + β 2 (nX ,neh ,T ) 20 (T ) X −X nX X −eh neh (53)

γ Here 20 (T ) denotes the background dephasing rate at given temperature T,

β β ≈ -1 2 β X − X is the exciton-exciton scattering parameter X − X 22 s cm and X −eh is the

90 exciton -eh -pair-scattering parameter. The two-dimensional exciton density nX and carrier density neh in the QW was estimated using relation

(−OD ) N ⋅ w ⋅ 1( − e X (eh ) ) ⋅ 1( − R) = hν X (eh ) nX (eh ) (54) π ⋅ r 2

Here N hν is the total number of photons of excitation pulse k1 , wX (eh ) gives the ratio of the number of photons that contribute to the exciton and the continuum absorption, respectively, R accounts for reflection losses at the sample surface. OD X (eh ) is the peak

optical density of exciton absorption lines and of the average optical density of the

exciton continuum, respectively. r is the 1 / e2 focus radius of the pulse on the sample.

≈ ⋅ 9 −2 With these parameters we estimate the exciton density to nX 1 10 cm and eh -pair

≈ ⋅ 9 −2 density to neh 4 10 cm for an average pulse power of 1 mW (pulse peak intensity of 2.1 MW/cm 2) at present excitation condition. Due to the weak overlap of the excitation

pulse with the barrier line any contribution of Xh excitons to the QW exciton density was neglected. The inset of figure 29 shows the dephasing rates that were extracted from the

FWM traces as a function of the electron-hole pair density. The dashed line reveals the

γ γ = -1 calculated rate 2 using a background rate of 20 (T ) 0.35 ps and a scattering

β = -1 -2 parameter of X −eh 90 s cm revealing that the exciton-eh -pair scattering starts to

2( d) < ⋅ 9 −2 saturate at densities below neh 5 10 cm .

β = ± -1 -2 The extracted value of X −eh 90 20 s cm is by a factor of 4 larger than

β X − X which is explained by stronger long-range Coulomb interaction between excitons

91 β β ≈ and free carriers. However, our investigations show a smaller ratio X −eh / X − X 4 as compared to measurements on bulk GaAs and on GaAs/InGaAs quantum wells

β β ≈ performed with ps pulses that reveal a ratio of X −eh / X − X 10 [86]. The observed

discrepancy is mainly attributed to different experimental conditions. While in ref. 87

β X − X accounts for the scattering rate of excitons with incoherent excitons (using a 20

β ps prepulse in a three-beam configuration) our scattering rate X − X obtained in a two- pulse configuration reflects the scattering between mainly coherent excitons [88]. Earlier investigations [89] demonstrated that the scattering process between coherent excitons is by a factor of ~3 more effective than a scattering with incoherent excitons therefore

β β explaining the reduced ratio X −eh /X − X in our studies compared to investigations by

Schultheis et al. [86].

5.2.2. Three-beam experiments

The importance of the eh -pair grating at pulse overlap is further supported by

three-beam experiments performed in (σ +σ +σ + ) configuration. Figure 30 shows the 11h

τ τ = signal trace as a function of variable delay 13 but at fixed delay 12 0 at k1 , k 2 and

2 k 3 excitation intensities of 1.3, 1.3 and 1.0 MW/cm , respectively. The center of the

τ pulse energy was set to 2.805 eV. For negative delay 13 < 0 (pulse k3 comes first)

τ essentially no signal is observed when delay 13 exceeds the exciton dephasing time

γ -1 τ γ -1 ( 2) . For positive delay 13 > ( 2) the diffracted signal intensity reveals a bi- exponential decay. The observed behavior is attributed to a nonlinear polarization

92 )3( ω τ > P + − ( , 13 )0 that is caused by the interaction of polarization k3 with both an k 3 k 2 k1

exciton grating (slow decaying component) and an eh -pair grating (first initial decay)

τ = generated at 12 0 .

τ = 10 ps 13

FWMsignal -1 0 1 delay τ [ps] 12 FWM signal 1 k - 2 k + 3 k

60 50 40 30 20 10 0 -10 delay τ [ps] 13

+ − τ Figure 30: FWM trace into k 3 k 2 k1 direction at the 11h exciton transition as a function of delay 13 σ +σ +σ + τ = for polarization configuration ( ) at 12 0 (open circles). The intensities of pulses k1 , k 2 and 2 k3 were 1.3, 1.3 and 1.0 MW/cm , respectively. The center of the pulse energy was set to 2.805 eV. Also shown are results of model calculations (full, dashed, dotted-dashed lines) as described in the text. The τ τ = inset shows the FWM trace as a function of delay 12 but for fixed 13 10 ps.

Since the eh -pair grating relaxes into an exciton grating the total signal intensity strongly

τ τ depends on delay 12 as shown in the inset of fig. 30. In this measurement delay 13 was

τ = τ kept at 13 10 ps but 12 was varied. Because of the broad spectrum of excited electron-hole-pairs no eh -grating is formed for delays much longer than the

+ − autocorrelation time between pulses k1 and k 2 . This explains the k 3 k 2 k1 signal decrease by a factor of >25 beyond pulse overlap.

For a quantitative analysis of the experimental data we model the FWM trace using the OBEs of a two level system that includes EID. We consider an enhanced EID

93 due to scattering at the eh -pair grating compared to EID at the created exciton grating.

We neglect all exciton transitions higher than the 11h transition and only consider

τ = incoherent interaction of polarization k3 with polarizations k 2 and k1 at delay 12 0 .

The Fourier transformed nonlinear polarization reads [88]:

  2 Nβ −  P )3( (ω, τ > )0 ∝ exp( iωτ )n exp( −γ τ ) + i X X  + X 13 13 X 0 1 13 ω − Ω ω − Ω 2  ( 21 ) ( 21 ) 

 2 Nβ −  n F(τ ) + iG (τ ) X X  + (55) eh 0 13 ω − Ω 13 ω − Ω 2  ( 21 ) ( 21 )    2 Nβ −  n K(τ ) + iL (τ ) X eh  eh 0 13 ω − Ω 13 ω − Ω 2  ( 21 ) ( 21 ) 

In equation (55) we use the abbreviations

Ω ≈ ω − γ − β + τ − β + τ 21 21 i 2 i X −X (nX 30 n X ( 13 )) i X −eh (neh 30 neh ( 13 ))

ω where 21 denotes the angular frequency between ground state and the 11h exciton

γ β β level, 2 is the exciton dephasing rate in the low density limit and X − X and X −eh

denote the exciton-exciton and exciton-eh-pair scattering parameter, respectively. nX 30

and neh 30 is the created exciton and eh-density, respectively, created at the arrival of k3 ,

τ τ nX ( 13 ) and neh ( 13 ) are the exciton and the eh-density, respectively, created by pulses

τ k 2 and k1 as a function of 13 .

The first term in the bracket of eq.(55) accounts for interactions of polarization

τ k3 with the exciton grating resonantly excited by pulses k 2 and k1 as a function of 13 ,

τ γ nX 0 being the exciton density created by pulses k 2 and k1 at 12 = 0 and 1 being the

combined rate of exciton recombination and grating diffusion. The first term in the brace

94 is due to phase space filling (PSF), the second term considers EID with N being the total number of available exciton systems involved. The second and third term in the bracket account for the interactions of polarization k3 with the exciton grating formed by the eh -

pair grating and with the relaxing eh -pair grating, respectively. neh 0 denotes the eh-pair

τ density created by pulses k 2 and k1 at 12 = 0. In the third term we use the same PSF nonlinearity for the eh -grating as for the exciton grating interaction but we introduce a 10

β β τ times higher scattering parameter X −eh as compared to X − X . Functions F( 13 ) and

τ G( 13 ) consider the formation and decay of the exciton grating that was formed by

τ τ relaxed eh-pairs, and K( 13 ) and L( 13 ) denote eh -pair relaxation functions.

For the formation of excitons from photoexcited electrons and holes we tentatively assume a geminate process where eh -pairs conserve their correlation during relaxation [90, 91] rather than a bimolecular formation which starts from uncorrelated electrons and holes in the conduction and valence band [92, 93]. This assumption is supported by three-beam experiments with varying k1 and k 2 pulse intensities up to 6.0

MW/cm 2 which reveal a constant eh -pair relaxation rate to form the exciton grating.

While this is an expected behavior for a geminate process the exciton formation rate in

the bimolecular case is expected to increase with increasing density of electrons and

holes. Detailed investigations applying higher intensities are presently in progress.

The eh-pair and exciton densities in the geminate process obey the differential equations

dn (t) eh = −γ n (t) (56) dt R eh

95 dn (t) X = γ n (t) − γ n (t) (57) dt R eh 1 X

For simplicity we neglect radiative and nonradiative losses of the eh -density neh (t ) and

assume that low temperatures prevent a thermal reactivation of excitons to eh -pairs in

γ eqs. (56) and (57). R is the geminate exciton formation rate. The solutions entering the

)3( ω τ > nonlinear polarization PX ( , 13 )0 in eq. (55) are:

γ F(τ ) = G(τ ) = R ()exp( −γ τ ) − exp( −γ τ ) (58) 13 13 γ − γ R 13 1 13 1 R

τ = τ = −γ τ K ( 13 ) L( 13 ) exp( R 13 ) (59)

To model these data of Fig. 30 with eq. (55) we used the exciton-eh -pair

β = −1 2 scattering parameter X −eh 90 s cm that was determined from the intensity dependent two-pulse experiments. Due to scattering with incoherent excitons we use a 10

β = −1 2 times lower exciton-exciton scattering parameter X −X 9 s cm was used in

accordance to [86].

Fig. 30 shows the model calculations using the geminate formation process where

dashed, dashed-doted, and full curves give the nonlinear polarization due to k 3

interaction with just the exciton grating, with the eh -pair grating only and with both

density gratings, respectively. The calculations show very good agreement to the

γ = −1 experimental data using the experimental determined recombination rate 1 01.0 ps ,

γ = −1 a geminate formation constant of R 2.0 ps and a total number of excitonic

oscillators of N = 7 ⋅10 10 . Deviations of the calculated curve with the experimental data

96 τ < within 13 1 ps are attributed to coherent signal contributions that have not been

considered in our model. The model calculations supply an intensity independent exciton

γ −1 ≈ formation time of R 0.5 ps that is by a factor of ~4 faster than obtained from GaAs

QWs revealing a formation time of < 20 ps [94]. The difference between these values are attributed to the enhanced LO-phonon interaction in more polar ZnSe as well as to a closer energy match between the exciton binding energy (~20 meV) and the LO-phonon energy (~32 meV) in the ZnSe QW. Both effects may lead to a faster relaxation of eh -

pairs in the continuum thus resulting in a faster exciton formation time.

97 5.3. Observation of exciton-LO phonon polarons in a 3 nm ZnSe QW

X X h l barrier PLE PLE intensity PL intensity,PL reflectivity

2.9 3.0 3.1 energy [eV]

Recently we performed two-beam degenerate FWM experiments on a 3 nm thick

ZnMgSSe/ZnSe QW using ~30 fs pulses. The sample consists of a 3 nm wide ZnSe QW sandwiched between two Zn 0.9 Mg 0.10 S0.16 Se 0.84 barriers of 30 nm thickness on the top and

60 nm at the bottom, with a 20 nm thick ZnSe buffer layer between the barrier and the

substrate. A similar structure with 4 nm multiple QWs and same barrier composition

revealed a heavy-hole exciton binding energy of ~ 33 meV as derived from two-photon

excitation (TPE) spectroscopy [77]. The full line in Fig. 31 shows the PL spectrum of the

3 nm ZnSe QW structure at 30 K revealing a strong 1s heavy-hole exciton X h emission

at 2.889 eV. The full width at half maximum (FWHM) of the emission line is ~4 meV

indicating some inhomogeneous broadening due to width fluctuations and weak

interdiffusion from the barriers. The weak signal at 2.858 eV is attributed to the X h -1LO

replica corresponding to a LO-phonon energy of ~31 meV that is close to the value of

98 31.7 meV known from bulk ZnSe [95].The additional broadening of the phonon replica with FWHM ~6 meV compared to the X h PL line and the slight variation of the LO- phonon center energy compared to the bulk value is attributed to disorder in the QW caused by the quaternary barriers. The PL spectrum further shows the energetic positions of the barrier excitons at ~3.1 eV. These energies are 80 meV higher as compared to the 4 nm QW sample used in earlier TPE measurements [77] indicating a higher S and Mg concentration and hence a stronger quantum confinement as nominally specified. The dotted-dashed curve shows the PLE spectrum of the 3 nm QW sample. The PLE signal which was detected at the maximum of the X h PL peak reveals a broad (~25 meV

FWHM) light-hole exciton X l transition at 2.931 eV that is separated by ~ 42 meV from

the X h emission. The large X l bandwidth indicates broadening due to X l coupling to the X h continuum and due to stronger penetration of the X l wavefunction into the disordered barriers.

τ =100 fs 2.86 12

2.88 XX X h

2.90

2.92 energy [eV] Xh+ LO X 2.94 l 0 1 2 3 delay τ [ps] intensity 12

Figure 32: Contour plot (logarithmic scale) of the FWM signal of a 3 nm ZnSe QW obtained in collinear ↑↑ τ ( ) configuration as a function of delay 12 between pulses k1 and k 2 with center energy 2.9 eV. The total 2 pulse intensity is 15 MW/cm . Also shown is the FWM spectrum at τ12 = 100 fs on a logarithmic scale.

99 ≥ Additional broadening is caused by n 2 X h transitions on the low energy side of the

X l band. These assignments are further supported by reflection measurements using white light emission from a halogen lamp (open circles) and a 30 fs laser pulse (dashed line) that was used in the FWM experiments. Only a small Stokes shift (~ 1 meV) of the

1s X h transition is observed, somewhat smaller than the 2.4 meV expected from the X h

linewidth of 4 meV [96].

Figure 32 shows the contour plot of the FWM measurements in (↑↑) configuration on a

τ logarithmic scale as a function of delay 12 between pulses k1 and k 2 with center energy

2.9 eV. The total intensity of the pulses was ~15.0 MW/cm 2. Also shown in Fig. 32 is the

FWM spectrum at τ12 = 100 fs on a logarithmic scale. Besides the photon echo of the inhomogeneously broadened Xh transition at 2.889 eV and the biexciton induced

transition XX the FWM signal reveals a spectrally broad (~12 meV at FWHM) and fast

decaying signal with maximum at 2.920 eV. The FWM spectrum does not exhibit a

spectrally resolved light-hole exciton signal Xl that is separated by ~42 meV from the Xh

transition. The weakness of the Xl FWM signal is attributed to a high scattering rate with

+ + electron-hole (eh )-pairs within the Xh continuum. The use of co-circular ( σ σ ) pulses in

the FWM experiments suppresses the biexciton induced transition XX but shows the same

Xh photon echo and spectrally broad feature with maximum signal energy at 2.920 meV as summarized in Fig. 33. The unexpected signal that is separated by ~31 meV from the

Xh transition is attributed to an exciton+1LO quasiparticle that is caused by a mixed coherent polaron mode.

100 Our assignment is based on the following arguments: As mentioned earlier a X h

= exciton binding energy of E1s 33 meV was observed in a similar 4 nm ZnSe QW sample. The value was deduced from the 1s to 2p exciton energy difference

∆ = 1s−2s 27 meV obtained from TPE measurements [77]. Due to the smaller QW size (3

nm) and higher confinement of the sample investigated in these FWM studies we expect

≥ ∆ ≤ an even higher Xh binding energy E1s 33 meV and a 1s-2s splitting energy of 1s−2s

29 meV. The strong Fröhlich interaction with a constant of α = 0.43 [81] and the high Xh

binding energy exceeding the LO-phonon energy generates a strong coupling between the

LO-phonon mode and the Xh exciton level. This strong coupling leads to the formation of a coherent mixed exciton+1LO polaron not unlike to the coherent vibronic progression of excitons in molecules [97] or in quantum dots [98] where the phonon coupling leads to sidebands in the FWM spectrum. Recent investigations have shown that stable mixed polarons can be realized in InAs quantum dots [99] where the energy splitting between electronic levels nearly matches the LO phonon energy. This stabilization process might also contribute in our QW sample where the expected 1s-2s Xh energy as well as transition energies of n > 2 exciton states lie close to LO-phonon energy. In contrast to stable polarons the coupling of higher order LO-phonons to the Xh continuum leads to a decomposition of the exciton+1LO-phonon polaron into a “free” phonon and scattered zero-phonon exciton in our sample. In addition the LO-phonon lifetime due to the decay into LA and TA phonons limits [100, 101, 102] the stability of the exciton+1LO-phonon quasi-particle.

101 X h "X " l XX X +LO k k h 1 2

σ+ σ+ FWM signal [normalized]

2.86 2.88 2.90 2.92 2.94 energy [eV]

+ + Figure 33: Normalized FWM spectra obtained in co-circular ( σ σ ), collinear ( ↑↑ ) and cross-linear ( ↑→ )

configuration, as labeled, at pulse delay τ12 = 0 and pulse center energy 2.9 eV. The total pulse intensity is 15 MW/cm 2. The spectra are offset for better comparison.

The statement of strong coupling between Xh excitons and LO-phonons is further supported by FWM experiments that were performed with pulses at a center energy of

2.915 eV that also cover the exciton+2LO energy. The used total excitation intensity of the pulses was ~35 MW/cm 2 in this case. The resulting FWM spectra are shown in Fig.

34 at different delay times as labeled. Besides the Xh exciton signal and mixed Xh+1LO

sideband faint structures at energies Xh+2LO and at Xh-LO become visible for positive

delay times. This observation is again attributed to the formation of coherent mixed

polaron modes. It should be noted that none of such LO sidebands have been observed in

previous quantum kinetic studies using III-V or II-VI bulk semiconductors [103, 56, 81,

98] or QW structures [104]. However, in none of those samples the exciton binding

energy exceeded the LO phonon energy. Therefore even in the case of bulk ZnSe [56] or

CdSe [104] with a coupling constant of α ≈ 0.4 the coupling is too weak to observe coherent polaron modes. Due to non-Markovian exciton LO-phonon scattering, however, clear phonon quantum beats between mixed states have been observed at the spectral position of the exciton transition shortly after the pulse excitation.

102 X -1LO X +1LO X +2LO τ [fs] h h h 12

+300 +200 +100

FWM intensity 0

-50 -100 2.88 2.91 2.94 2.97 energy [eV]

The FWM traces at the spectral position of the Xh exciton and at the mixed polaron energies Xh+1LO and Xh+2LO of our QW sample are displayed in Fig. 35 on a

logarithmic scale. Also shown in the inset are the Fourier transformed spectra at energy

positions Xh and Xh+1LO, respectively. The Fourier transformed spectra reveal weak (but well above noise level) signal maxima around 31 meV as expected from the FWM spectra and in agreement to studies in bulk ZnSe [56]. The FFT signal broadening on the high energy side might indicate the contribution of continuum-LO-phonon coupling with

ω = ω + = a renormalized beat frequency LO LO 1( me / mh ) . With me .0 145 m0 and

= ω ≈ mh 8.0 m0 [3] the renormalized frequency amounts to LO 37 meV . Continuum-

LO-phonon beats have been observed in GaAs bulk material [103, 56] with weak

coupling constant α = 0.06 and indirectly in CdSe bulk samples where beat frequencies

between the bulk and the renormalized value have been found [104]. Frequencies larger

than ~40 meV may originate from weak heavy-hole-light-hole Xh- X l exciton beats. The contribution of the 2s exciton to the Xh+1LO polaron signal is estimated according

103 ∝ 2 ⋅ 2 relation I FWM o1s o2s where o1s and o2s are the oscillator strength of the 1s and 2s

= exciton, respectively. Due to the small ratio of 1s/2s oscillator strength that is o1s /o2s

= 1/8 for bulk material and o1s /o2s 1/27 in pure 2 dimensional (2D) QWs we expect a more than 100 times lower 2s Xh FWM signal compared to the 1s signal at τ12 = 0 in our

QW sample. Accordingly we anticipate that the Xh+1LO polaron signal exceeds the 2s

exciton signal but 2s transitions may contribute to the broadening on the low energy side

of the Xh+1LO signal. (See figs. 31 to 34).

Xh X X +1LO h h FFT intensity

10 20 30 40 50 60 70 ∆E [meV] X + 1LO h FWM FWM intensity

Xh+ 2LO

0 1 2 3 τ delay 12 [ps]

Figure 35: FWM traces at the spectral position of the heavy-hole X h exciton and at mixed polaron energies Xh+1LO and X h+2LO on a logarithmic scale. The pulse center energy is 2.915 eV; the total pulse intensity 2 is 35 MW/cm . The inset shows the Fourier transformed spectra of the FWM traces at positions X h and Xh+1LO, respectively.

The photon echo (PE) at position X reveals an exciton dephasing time of T = 3.1 h X h

ps which is predominantly limited by exciton-carrier scattering at applied high excitation

densities. The FWM traces at the energy position Xh+1LO and Xh+2LO also show the

character of a PE but reveal a faster dephasing time of T + ≈ 700 fs and X h 1LO

T + ≈ 900 fs. The enhanced dephasing of the exciton+1LO quasi-particle is X h 2LO attributed to the decay of the coherent polaron into a 1s exciton state and free LO-

104 phonons. Due to the vicinity of the 2s transition energy to the 1LO-phonon energy an effective decomposition rate of the Xh+1LO superposition state is expected. This nearly

resonant condition is not fulfilled for the Xh+2LO quasi-particle, which might explain the

slightly longer coherence time for this polaron mode. Also the inhomogeneous

broadening of LO-phonons energies due to disorder and their dispersion in the k-space as well as the disintegration of the LO phonon into acoustic and other phonons may contribute to the faster decay of the superposition states compared to the Xh transition.

While the later case is probably of minor importance since most of the LO-phonon

lifetimes in III-V [100] or II-VI materials [101] exceed several picoseconds

inhomogeneous broadening of LO-phonons may affect the dephasing of superposition

states due to destructive interference of LO-phonon wave-packets [105] involved.

τ = 15 fs 2.86 12 XX 2.88 X h

2.90

energy [eV] energy 2.92 "X " l 2.94 -0.5 0.0 0.5 1.0 1.5 delay τ [ps] 12 intensity

Figure 36: Contour plot (logarithmic scale) of the FWM signal obtained in cross-linear ( ↑→ ) configuration τ as a function of delay 12 between pulses k1 and k 2 with center energy 2.9 eV. The total pulse intensity 2 is 15 MW/cm . Also shown is the FWM spectrum at τ12 = 15 fs on a logarithmic scale.

Figure 36 shows the contour plot of the FWM measurements for cross-circular ( ↑→)

τ fields on a logarithmic scale as a function of delay 12 between pulses k1 and k 2 with center energy 2.9 eV. The total intensity of the pulses was 15 MW/cm 2. Also shown in

105 Fig. 36 is the FWM spectrum at τ12 = 15 fs on a logarithmic scale. As expected the

biexciton induced transition XX gains in importance due to the lack of excitation induced

dephasing (EID) in this configuration. The Xh FWM signal is reduced by a factor of ~8 compared to (↑↑) fields. Also the Xh+1LO mixed polaron sideband is strongly reduced

while a new band with maximum at 2.925 eV appears that is shifted by ~5 meV with

respect to the Xh+1LO maximum observed in (↑↑) configuration (see Fig. 33). This band

is attributed to localized “ Xl” transitions that lie energetically below the Xh continuum

and are therefore less affected by rapid scattering processes with eh-pairs. This

≈ interpretation suggests a 1s Xh binding energy of E1s 36 meV and a 1s-2s splitting

∆ ≈ energy of 1s−2s 29 meV using a fractional dimension approach [7] which are close to values that we expect from the QW structure. Additional FWM measurements in ( ↑→) configurations with pulse center energy 2.915 eV show the same dynamical behavior, in particular, no 1LO- or 2LO-sidebands a visible in the FWM spectrum.

XX X h

"X " l FFT intensity

0 10 20 30 40 50 60 "X " ∆ l E [meV] FWM intensity FWM

0.0 0.5 1.0 1.5 delay τ [ps] 12

106 Figure 37 finally shows the FWM traces in ( ↑→) configuration at the spectral

position of the Xh and light-hole related “ Xl” transition. As in the collinear ( ↑↑)

configuration the inhomogeneous broadened transition states show a PE decay. The Xh

dephasing time results to T ≈ 1 ps which is slightly longer as the “ X ” dephasing that is X h l determined to T ≈ 800 fs . The shorter dephasing time of the X transition in ( ↑→) Xl h compared to ( ↑↑) configuration is attributed to an additional decay that is caused by the inhomogeneous broadening of bound and unbound biexciton states [106]. The Fourier transformed signal at the Xh energy shown in the inset of Fig. 37 reveals a frequency of

11 meV due to a beating between biexciton induced XX and heavy-hole Xh transitions as

well as a broad band with center frequency at ~39 meV which is predominantly attributed

to the Xh-Xl beating.

107 5.4. Phase coherent photorefractive (PCP) effect

5.4.1. Two-beam FWM measurements; Observation of PCP

I/I 0 p /I

s 1

1/10

1/40 ~ I 2 p signal[normalized] -4 -2 0 2 4 delay τ [ps] 12 diffraction efficiency I

10 100 1000 excitation intensity [kWcm -2 ]

Figure 38: Dependence of the diffraction efficiency at the spectral position of the 11H Xh exciton transition recorded at 55 K. The dashed line shows a fitted curve as described in the text; the solid line shows an 2 I p dependence. The inset demonstrates intensity dependent FWM traces at different k1 and k 2 pulse -2 intensities as labeled. ( I0 = 800 kWcm )

An all-optical PR effect (PRE) in ZnSe quantum wells (QWs) using ultra-short light pulses resonant to the excitonic transition has been observed while performing two- beam degenerate FWM experiment at low intensities. Since the phase and polarization of the incident pulses is preserved by the coherence of excitons the PRE can be observed for pulses that do not overlap in time. Of particular interest is the possibility to erase the phase coherent photorefractivity (PCP) at temporal pulse overlap and to store one logic 0 bit and three different logic 1 bits using various circular pulse polarizations during the exciton coherence time. The different logic bits can be distinguished by the diffracted beam intensities as well as by their optical field polarizations. The additional degree of phase coherence in these novel PCP structures may open new design possibilities for optical data storage and computation.

108 The inset of Fig. 38 shows intensity dependent two-beam FWM traces in

+ + (σ σ ) configuration at the spectral position of the X h heavy-hole exciton at 2.806 eV.

The center of the 22 meV broad excitation pulse was set to 2.804 eV in order to predominantly excite Xh excitons but to avoid continuum contributions. At intensity I0 =

800 kWcm -2 (corresponding to an average power of 0.45 mW) the exciton resonant signal agrees with a χ (3) nonlinear optical FWM process as described by extended optical

Bloch equations for a homogeneously broadened system where the exciton lifetime T1

significantly exceeds the exciton dephasing time T2 .

τ In such a case at positive delay time 12 the exponentially decay of the FWM

τ τ signal with 12 is given by 2/ T2 (with dephasing time T2 ~ 1.55 ps), while at negative 12

when the FWM is due to excitation induced dephasing the rate is twice as large (4/ T2)

[107]. With decreasing pulse intensity, the signal becomes symmetric; the decay rate for

negative delay times approaches a value 2/ T2, and a pronounced dip at temporal pulse

τ overlap at 12 ~ 0 appears. Additionally, the maximum signal intensity Is is not scaling

3 χ (3) with the excitation intensity I p like I p as expected for a FWM process. This is documented in Fig. 38, where the diffraction efficiency I s I/ p is given as a function

2 of I p . The solid line shows the expected I p dependence while the dashed line shows a fit

= + ⋅ 2 η ξ of the experimental data by I s I/ p η ξ I p with parameters and . While the quadratic power dependence of a χ (3) FWM process is observed at intensities above 200 kWcm -2, the diffraction efficiency is essentially constant at lower intensities, with

109 η /ξ = 3.2 ×10 5 W2/cm 4 . A calibration of the used detector system revealed an estimate

value of the diffraction efficiency η ≈ 3×10 −4 .

n ne after Pulse k1

Excitons n X CB after Pulse e- k1+ k2 VB

e- grating after ~1ps substrate well time barrier barrier x z Figure 39: Schematic diagram (left) shows the steps for formation of the electron grating in the quantum well after the decay of the exciton grating. The right figure shows the transfer of electron from the GaAs substrate to the quantum well.

The experimental findings show that the diffracted signal at low excitation

intensities is not generated by a χ (3) FWM process. Instead we attribute it to an exciton

resonant PRE caused by a long living electron density grating that is written into the QW

by pulses k1 and k 2 . We propose the following grating formation process (see fig. 39):

The excess electrons, as evidenced by previous studies [108], are optically excited in the

GaAs substrate and subsequently captured in the QW. An exciton grating is generated by interference of coherent exciton polarizations created by pulses k1 and k 2 . The electrons are redistributed by the exciton grating due to Coulomb interaction and Pauli blocking.

After about the exciton lifetime (~100 ps) the exciton density grating vanishes by recombination, while the long living electron grating that is stabilized by localized holes at the ZnSe/GaAs interface remains in the QW. Longitudinal space charge fields within the QW and the GaAs/ZnSe interface cause a periodic modulation of the optical constants

110 that leads to a diffraction of exciton polarisation P into direction 2k − k . In addition k2 2 1

to the PR signal caused by exciton energy modulation by the quantum confined Stark

effect the spatial modulation of the electron density leads to a PR signal via the

modulation of the exciton dephasing time by excitation induced dephasing (EID).

The efficiency of the PRE is proportional to the modulation of the electron

density within the QW. The nearly constant diffraction efficiency η in Fig. 38 implies

that the electron density modulation is nearly independent of the excitation intensity.

Since the exciton density grating is proportional to the excitation intensity the decay rate

of the formed electron grating must also scale proportional to the excitation intensity.

This behaviour can be actually expected considering the electron density formation

process: The electron capture rate into the QW is proportional to the electron excitation

rate in the GaAs substrate, which in turn is proportional to the excitation intensity. In

equilibrium, the electron escape / tunnelling rate out of the QW via the vertical space

charge field must be equal to the capture rate, and thus the electron grating decay rate is

proportional to the excitation intensity (assuming a nearly constant electron density, as

discussed later).

The observed delay time dependence of the diffracted signal (see inset of Fig. 38)

demonstrates that the PRE is proportional to the exciton density modulation, which

∝ − τ τ exponentially decreases exp( 2 12 / T2 ) with delay 12 between the writing pulses

τ ≈ according to the exciton dephasing time. The signal drop during pulse overlap at 12 0

is unexpected in first place. However, during pulse overlap, also the excitation intensity

of the GaAs substrate is spatially modulated, giving rise to a second contribution to the

111 electron density grating, which increases the electron density in the region of larger exciton density, and thus counteracts the exciton density grating effect.

Xh Xh σ+ σ+ FWM signal

4 3 2 1 0 -1 -2-3-4 τ delay 21 [ps]

-6

FWM-signal -4 -2 s] 0 [p 2.80 2 τ 12 2.81 4 lay 6 de ene 2.82 rgy [eV]

− τ Figure 40: Spectrally resolved PR signal into 2k 2 k1 direction as a function of delay 12 recorded at a lattice temperature of 55 K. The total intensity of the co-circular polarized pulses is 20 kWcm -2. The inset shows the calculated trace at the Xh heavy-hole exciton transition as described in the text (full line) in comparison to the experimentally observed trace (dashed line).

− Figure 40 shows the spectrally resolved photorefractive (PR) signal in 2k 2 k1

τ σ +σ + direction as a function of delay 12 for ( ) polarized fields at a temperature of 55

K. The center of the 22 meV broad excitation pulse was set to 2.808 eV, in order to avoid

continuum contributions and predominantly excite the Xh heavy-hole exciton at 2.806 eV

and weakly excite the light-hole X l exciton transition at 2.821 eV. The total excitation

-2 intensity of pulses k1 and k 2 was 20 kWcm in these experiments (corresponding to a

total average power of 14 µW). As described above the signal slope approaches the value

γ γ τ 2 2 (with 2 being the exciton dephasing rate) for negative ( 12 < 0) and positive delay

τ times ( 12 > 0). Furthermore the signal trace reveals a pronounced dip at temporal pulse

τ overlap 12 = 0. In addition quantum beats between heavy- and light-hole excitons with a

112 ∆τ ≈ ∆ = ∆τ ≈ period of 290 fs (corresponding to an energy splitting of ∆∆Ehl h / ∆∆ 15 meV) are observed in the PR signal traces.

For a quantitative description the PR response obtained in two pulse configuration

is modeled using the optical Bloch equations (OBE) of a five-level system for δ-shaped

laser pulses. The five-level model with the ground level 1 initially populated takes into

+ − account the nature of σ and σ Xh excitons (levels 2 and 3 ) and Xl excitons (levels

4 and 5 ). After Fourier transformation the diffracted polarization into direction

− 2k 2 k1 reads:

5 ⋅ j1( j1 e2 ) ω τ ∝ − τ + τ − ωτ P2 ( , 12 ) ( ak −k ( 12 ) bk −k ( 12 ))exp ( trep /Tg ) ∑ exp (i 12 ) 60( ) 2 1 2 1 Ω −ω j=2 ( j1 )

Function a (τ ) describes the density of the exciton grating that is k2 −k1 12 responsible for the subsequent formation of an electron grating in regions of low exciton density, given by:

5 τ = − ⋅ * ⋅ * Ω τ ak −k ( 12 ) a θ( τ12 )∑( j1 e2 )( j1 e1 )exp(i j1 12 ) 2 1 j=2 61( ) 5 + ⋅ * ⋅ * Ω* τ a θ(τ12 )∑( j1 e2 )( j1 e1 )exp(i 12 ) j1 j=2

σ + σ − Unit vectors e1 and e2 denote the polarization unit vectors of the and

± = −0.5 polarized laser pulses k1 and k 2 with σ 2 1( ,m )i in Jones notation. The optical

= µ + = µ + = µ − matrix elements j1 are given by 12 12 σ , 15 15 σ and 13 13 σ ,

= µ − µ 14 14 σ where j1 accounts for the magnitude of the dipole transitions.

113 Ω = ω − γ ω Furthermore we have used the abbreviations 1j 1j i 2 where the 1j denote the

+ − angular frequencies between ground state 1 and the σ and σ exciton levels 2 , 3 ,

4 and 5 , respectively. The empirical parameter a accounts for the efficiency of the

= PR effect. trep 12 5. ns is the pulse repetition rate. Time constant Tg gives the lifetime of

the electron grating that is determined by lateral diffusion and electron escape,

considering that polarization P is diffracted from the partly decayed “old” k − k k2 2 1

grating that was created in the previous pulse cycles.

Function b − (τ ) in eq. (60) describes the electron density grating that is k 2 k1 12 created during pulse overlap given by:

* ( ) b − (τ ) = b e ⋅e exp( iω τ ) E (t −τ )E (t)dt 62( ) k 2 k1 12 2 1 p 12 ∫ k2 12 k1

In equations (62) the integral gives the field autocorrelation of sech(t) shaped pulses with temporal full width half maximum (FWHM) of 1.49 ∆t where ∆t is the

ω measured intensity FWMH of the incident laser pulses. p is the center frequency of the pulse, b is an adjustable parameter in order to obtain optimum agreement with the

τ ≈ π observed signal at 12 0 . The minus sign in eq. (60) considers the shift of this

electron grating with respect to the formed exciton grating.

The inset in Fig. 40 displays the numerical calculation at the spectral position of

the Xh heavy-hole exciton (solid line). The energetic position of the exciton transition was

taken from the experimental spectra, the magnitude of the ground to heavy-hole dipole

µ = µ matrix elements 21 31 and that of the ground to light-hole exciton dipole matrix

µ = µ elements 41 51 were set to 1 and 0.57, respectively, according to their relative

114 µ 2 µ 2 oscillator strength-ratios 21 : 41 of 3:1 given by their different valence band Bloch

functions. In the calculation we further considered the spectral intensity of the pulse at

the Xh and Xl transition energy. From the signal trace we evaluated the exciton dephasing time to T2 = 1.55 ps. For the electron grating lifetime Tg we used 10 ms, for the temporal

FWHM of the excitation pulses we applied ∆t = 90 fs, and parameter b was adjusted to

0.6. The overall agreement between the model calculations and the experimental data is

good.

5.4.2. Three-beam FWM measurements of PCP effect

To confirm the above interpretation we performed three-pulse experiments with

τ τ = fixed 12 as function of 13 using circular polarized pulses of peak intensities I1 48 ,

= = −2 I2 32 , and I3 70 kWcm . In order to avoid the excitation grating in the GaAs

τ ≈ + − substrate during pulse overlap, we chose 12 200 fs. The signal in k 3 k 2 k1

+ + + + − + − − + direction resonant to the Xh transition in (σ σ σ ) , (σ σ σ ) and (σ σ σ )

τ configuration is displayed in Fig. 41 as a function of delay time 13 . (The scale in Fig. 41

− − + is the same for all sub-plots.) In (σ σ σ ) configuration we observe a diffracted signal

τ σ +σ −σ + that is nearly independent of delay 13 . In ( ) configuration an almost

τ = symmetrical signal with pronounced signal dip at 13 0 appears, similar to the signal

observed in the two-pulse experiments (Fig. 40). Finally, in (σ +σ +σ + ) configurations

τ both the 13 independent signal and the symmetrical signal coexist. No signal is observed

− + + for the (σ σ σ ) configuration.

115 2.812 σ+σ+σ+ 2.808

+ - +

signal σ σ σ 1 k - energy [eV] energy

2 2.804 k +

3 - - +

k σ σ σ 8 4 0 -4 -8 8 4 0 -4 -8 τ delay τ [ps] delay [ps] signal 13 13

+ − Figure 41: Traces of the diffracted k 3 k 2 k1 PR signal at the Xh exciton transition for polarization + + + + − + − − + configurations (σ σ σ ) , (σ σ σ ) and (σ σ σ ) recorded at 55 K. The inset shows the spectrally + + + resolved contour plot for polarization configuration (σ σ σ ) . In addition the PR signal spectrum at delay τ = 13 100 fs (solid line) and the spectrum of the 90 fs excitation pulse (dashed line) is displayed.

All experimental results are in agreement with the interpretation as PCP caused by the

σ −σ −σ + formation of an electron grating. In ( ) configuration exciton polarizations k1

and k 2 create an exciton density grating leading to the electron density grating, of which

τ > pulse k 3 is diffracted. For 13 0 pulse k 3 is diffracted from a “refreshed” electron

τ < grating while for 13 0 it interacts with an “old” electron grating that was induced by pulses k1 and k 2 of earlier repetitions. The almost equal signal strength for positive and negative delay times indicates an electron grating lifetime that is more than 100 times longer than the pulse repetition period of 12.5 ns. In (σ +σ −σ + ) configuration an exciton

density grating is formed between pulses k1 and k 3 , and pulse k 2 is diffracted of the resulting electron grating. Accordingly, the FWM intensity decays

∝ − τ σ −σ +σ + − exponentially exp( 2 13 / T2 ) . In ( ) configuration, neither k 2 k1 nor

− k 3 k1 exciton gratings are created, and accordingly no signal is observed. In

116 (σ +σ +σ + ) configuration both mentioned exciton gratings are formed, and the signal

− consists of two contributions: Pulse k 2 is diffracted of the k 3 k1 grating, and Pulse k 3

− is diffracted of the k 2 k1 grating. Since they have a time difference

− = τ −τ t3 t 2 13 12 , the interference of the two signals leads to spectral oscillations of the

diffracted intensity (see contour plot), which shows that the two diffracted polarizations

τ ≈ are phase correlated. The inset shows the PRE signal spectrum at 13 100 fs (solid line) and the spectrum of the used 90 fs excitation pulse (dashed line) demonstrating that the observed effect is resonant to the quantum well exciton and is thus not due to a non- resonant PRE in the substrate.

For a quantitative description we have modeled the PCP in three pulse

configuration using the optical Bloch equations of the exciton system for δ-shaped laser pulses. The 3-level model consists of the initially populated ground state 0 and two

+ − + − optically active Xh exciton states of σ and σ polarizations (levels σ and σ ).

ω τ τ After Fourier transformation the diffracted polarizations P2 ( , 13 , 12 ) and

ω τ τ + − P3 ( , 13 , 12 ) into direction k 3 k 2 k1 read:

⋅ ⋅ + (+ e2(3) ) − (− e2(3) ) ω τ τ ∝ τ τ + ωτ P2(3) ( , 13 , 12 ) ak −k ( 13(2) ) n e ( 13(2) )   exp( i 12(3) ) (63) 3(2) 1 Ω −ω Ω − ω  ( + ) ( − ) 

Functions a − (τ ) describe exciton gratings that are responsible for the subsequent k 3(2) k1 12(3)

formation of the PR electron gratings given by:

117 a − (τ ) = k )2(3 k1 13(2) * * * * − ( ⋅ ⋅ Ω τ + ⋅ ⋅ Ω τ ) (64) a )2(3 θ( τ13(2) () + e3(2) )( + e1 )exp(i + 13(2) ) (− e3(2) )( − e1 )exp(i − 13(2) ) + ( ⋅ * ⋅ * Ω*τ + ⋅ * ⋅ * Ω *τ ) a )2(3 θ(τ13(2) () + e3(2) )( + e1 )exp(i + 13(2) ) (− e3(2) )( − e1 )exp(i − 13(2) )

In equations (63) and (64), e1 , e2 and e3 denote the polarization unit vectors of the pulses

± −0.5 with σ = 2 1( ,m )i in Jones notation. The optical matrix elements + and − are given

+ − by + = µ σ and − = µ σ where µ accounts for the magnitude of the dipole transitions.

Ω = ω − Ω = ω − ω Furthermore we have used the abbreviations + + /i T2 and − − i /T2 where +

+ − and ω− denote the angular frequencies between ground state 0 and the σ and σ

exciton levels, respectively. The empirical parameters a3 and a2 account for the efficiency of the PCP.

In our model we also consider that the grating contrast ne decreases when the laser pulses overlap according to

n (τ ) = (1− c E (t −τ ) E (t)dt ) (65) e 12 (3) ∫ k3 12 (3) k1

In eq. (65) the integral gives the field autocorrelation of sech(t) shaped pulses with

temporal full width half maximum (FWHM) of 1.49 ∆t where ∆t is the intensity FWHM

of the incident laser pulses. c is an adjustable parameter to obtain agreement with the

τ ≈ observed signal drop at 13 0 .

Fig. 42 displays numerical calculations for the configurations (σ +σ +σ + ) ,

− − + (σ +σ −σ + ) and (σ σ σ ) . (The scale in Fig. 42 is the same for all sub-plots.) From the signal trace we evaluated the exciton dephasing time to T2 = 1.55 ps. The PR efficiency

∆ τ ratio a2/a 3 was set to 0.38/1 for the temporal FWHM we applied t = 90 fs, 12 was set

118 to 200 fs and parameter c was adjusted to 0.7. The agreement between calculated values

− − + and experimental data for (σ +σ −σ + ) and (σ σ σ ) configuration is good. The higher

PR efficiency in (σ +σ −σ + ) is attributed to the larger grating period larger grating period

µ − of the generated electron grating (the grating periods are 7 and 3.5 m for the k 3 k1

− and k 2 k1 grating, respectively) when considering the effect of electron diffusion in

the QW. For configuration (σ +σ +σ + ) the emerging polarization is the coherent

ω τ τ + ω τ τ superposition P2 ( , 12 , 13 ) P3 ( , 12 , 13 ) .

2.812

σ+σ+σ+ 2.808

signal + - + 1 σ σ σ k energy energy [eV]

- - 2.804 2 k + + 3 k σ-σ-σ+ 8 4 0 -4 -8 8 4 0 -4 -8 delay τ [ps] signal delay τ [ps] 13 13

+ − Figure 42: Calculated traces of the diffracted k3 k 2 k1 PR signal at the Xh exciton transition for + + + + − + − − + polarization configurations (σ σ σ ) , (σ σ σ ) and (σ σ σ ) for the same conditions as in Fig. 47. + + + The inset shows the calculated spectrally resolved contour plot for polarization configuration (σ σ σ ) τ = and the PR signal spectrum at delay 13 100 fs.

The higher signal strength in the model calculations compared to experimental

− − data in this configuration indicates that the gratings k 3 k1and k 2 k1 are partly erasing each other, an effect that is appreciable but beyond the scope of the presented

τ = model. Slight asymmetries of the experimental signal relative to 13 0 , not reproduced

119 τ < by the calculations, are attributed to a stronger diffraction of pulse k 2 for 13 0 by the

− refreshed k 3 k1 grating. Further discrepancies are attributed to the partial loss of coherence between polarizations P2 and P3 due to experimental instabilities. In addition

+ different scattering cross sections of σ excitons with spin-up or spin-down spin- polarised electron gratings (created by the optically excited spin-polarized exciton grating) also lead to efficiency changes in the PRE signal.

Fig. 42 shows the calculated contour plot of the diffracted polarization for

σ +σ +σ + τ ≈ configuration ( ) . The signal dip at 13 0 as well as the slight asymmetry due

τ ≈ to the time difference of 12 200 fs is well reproduced However, the calculated plot

reveals more pronounced spectral oscillations than the experimental data. Also here

observed partial loss of coherence is explained by experimental instabilities.

5.4.3. Spectral dependence of the PCP effect

Xh excitation [eV] σ+σ+

2.785

2.793

2.801

2.809 PRsignal [normalized]

2.817 -4 -2 0 2 4 τ delay 12 [ps]

− Figure 43: Traces of the diffracted 2k 2 k1 PR signal at the Xh heavy-hole exciton transition as a function τ of delay 12 recorded at a lattice temperature of 55 K and at different excitation energies as labeled. The = = −2 pulse intensities of co-circular polarized pulses k1 and k 2 are I1 80 , and I2 40 kWcm , respectively.

120 − Figure 43 shows the normalized PR signal traces in 2k 2 k1 direction as a

τ σ +σ + function of delay 12 for ( ) polarized fields at a temperature of 55 K. The center

energy of the excitation pulses was changed from 2.785 to 2.817 eV in 8 meV steps. In

these experiments the excitation intensity of pulses k1 and of k 2 was set to 80 and 40 kWcm -2, respectively, (corresponding to a total average power of 77 µW). While the signal trace excited at 2.809 eV shows a similar behavior as in Fig. 1 the traces at lower excitation energies reveal no Xh - Xl beating and possess higher decay rates. In addition

τ > the shape of the PR signals becomes asymmetric with faster decay for 12 0 and slower

τ < decay for 12 0 . In contrast the PR signal excited at highest energy (2.817 eV) shows a

lower decay rate for positive delay but a higher rate for negative delay. Furthermore the

signal dip starts to fill up at excitation energies above 2.801 eV.

excitation [eV] 2.785

2.793

2.801

2.809 PR signal [normalized] Xl 2.817 2.78 2.80 2.82 energy [eV]

121 τ Figure 44 depicts the PR signal spectra for a positive delay 12 = 500 fs obtained at

different excitation energies as labeled. In addition the spectra of excitation pulses are

given as dashed curves for comparison. While both heavy Xh and light hole excitons Xl

are excited at 2.817 and 2.809 eV all other spectra are dominated by the Xh exciton

explaining the lack of quantum beats in the signal traces below 2.809 eV (see Fig. 2).

Furthermore the peak position of the heavy hole exciton Xh is shifted to lower energies

when the ZnSe QW is excited above 2.801 eV.

Figure 45 (a) represents the relative PR signal intensities for the different excitation energies revealing a maximum signal intensity at an excitation energy of 2.801 eV. Fig. 45 (b) summarizes the energy shift of the heavy-hole exciton transition and the exciton decay rates as a function of the excitation energy.

2.78 2.79 2.80 2.81 2.82 a) 3

PR intensity 1

3 3 γ 2 [meV] 2 2

1 1 redshift [meV] b) 0 0 2.78 2.79 2.80 2.81 2.82 excitation [eV]

τ γ Figure 45: PR signal intensities (a) and energy shift at 12 = 500 fs as well as exciton dephasing rates 2 of

the Xh heavy-hole exciton transition (b) as a function of the excitation energy.

The observed dependencies clearly indicate that the resonant diffraction is caused both by a periodic modulation of the exciton transition energy, usually expected from QCSE, as

122 well as by a periodic modulation of the exciton dephasing time also known as excitation induced dephasing (EID).

According to our assumption that the resonant PCP effect is attributed to an electron grating created by captured QW electrons with density ne we conclude from

figures 28 to 30 that the electron density ne increases with decreasing excitation energy.

The increase of ne is attributed to an increased transparency of the ZnSe buffer layer

(excitation below band-gap) on top of the GaAs substrate as well as to a decreased density of excited QW excitons. The lower exciton density leads to reduced electron escape rate due to Auger process that are caused by the recombination of excitons. The increased captured electron density leads to an enhanced exciton-electron scattering that increases the exciton dephasing rate γ2 according to

γ = γ + β 2(nX ,ne,T) 2(nX ,T) Xe ne (66)

β with Xe being the exciton-electron scattering parameter and ne being the electron density. Expression

β γ (n T), = γ + β T + LO + β n (67) 2 X 20 ac − XX X exp (ELO (/ kBT )) 1 describes the exciton dephasing rate extrapolated to zero electron density with

γ β background dephasing rate 20 , acoustic and LO-phonon scattering parameters ac and

β β LO , respectively, and T being the lattice temperature. Furthermore XX is the exciton-

exciton scattering parameter and nX is the exciton density. To calculate the static space

charge fields we estimated the electron density ne using an acoustic phonon scattering

β = ⋅ 8 −1 −1 parameter ac 7.6 10 s K [89], a LO-phonon scattering parameter

123 β = ⋅ 13 −1 = LO 8.3 10 s [89], the LO-phonon energy ELO 31 .6 meV and an exciton-

β = −1 2 exciton scattering rate of XX 22 s cm [88]. From earlier investigations [48] we

≈ ⋅ 9 −2 −2 estimated an exciton density of nX 1.5 10 cm at 80 kWcm pulse intensity.

Recently performed intensity dependent FWM measurements on the same quantum well using 30 fs pulses reveal an exciton-electron-hole pair scattering parameter of

β ≈ ± −1 2 e−hX 70 20 s cm . If we assume that the scattering due to holes is significantly smaller than due to electrons [109] and also assume a background dephasing rate of

γ ≈ 20 0 we can estimate an (upper limit) electron density ranging from

≈ ⋅ 10 −2 ≈ ⋅ 10 −2 ne 7.0 10 cm at an excitation energy of 2.817 eV to ne 8.2 10 cm at an

excitation energy of 2.785 eV.

The increased electron density leads to an increasing electric field E between

quantum well electrons and positive charges at the barrier/substrate interface according to

= σ εε σ = E / 0 with surface charge density ne e , e being the electron charge and static

dielectric constant ε = 9. The maximum E field between the quantum well surface that is

− adjacent to the barrier substrate interface calculates to E ≈ 6 ⋅10 3 Vcm 1. Such fields

lead to a noticeable tilt of the Zn 0.94 Mg 0.06 Se barrier (with ~20 meV conduction band

offset [78]) and the ZnSe buffer layer band alignments hence reducing the effective

thickness of the barrier that decreases the electron and hole confinement energies due to

increased penetration of the wavefunctions into the barrier. The reduction of confinement

energies explains the observed redshift of the Xh exciton transition with increasing

density ne .

124 τ 4 12 < 0 τ > 0 12 2 X

redshift [meV] 0 h 0 1 2 3 10 -2 ne [x10 cm ] excitation [eV] 2.785

2.793

2.801

PR signal [normalized] 2.809

2.817 2.800 2.805 2.810 energy [eV]

− τ > Figure 46: Spectra of the diffracted 2k 2 k1 PR signal for positive 12 0 (solid line) and negative delay τ < 12 0 (dashed line) recorded at a lattice temperature of 55 K and at different excitation energies as

labeled. The inset shows the obtained redshift of the Xh heavy-hole exciton transition as a function of the captured electron density ne evaluated from eq. (66). The used pulse intensities are the same as in Fig. 43.

Figure 46 shows detailed PR spectra of the Xh signal for positive (solid lines) and

negative (dashed lines) pulse delay. The inset in Fig. 46 illustrates the observed redshift

of the Xh exciton PR signal as a function of the electron density ne derived from eq. (66).

Since the intensity of pulse k1 is by a factor of 2 higher than that of pulse k 2 the

electron densitiy ne differs for positive and negative delay leading to a stronger redshift

τ of Xh PCP signal for 12 > 0. Because the field lines start in the quantum well rather than they pass the QW the redshift of the exciton transition versus ne does not show a

quadratic dependence as expected from the QCSE. In addition, as mentioned earlier, EID

also efficiently contributes to the observed coherent PR diffraction.

At excitation energies higher than 2.809 eV the reduced transparency of the ZnSe

QW and of the ZnSe buffer as well as an increased exciton density nX lower the

captured electron density ne leading to a decreasing of PR efficiency. Hence the

125 otherwise weak χ(3) FWM signal process becomes the dominant contribution in the

τ ≈ observed signal also explaining the filling of the signal dip at 12 0 and the faster decay

τ for negative delay compared to 12 > 0.

5.4.4. Temperature dependence of the PCP effect

-2 Figure 47 shows the PR signal at pulse intensities of 40 and 30 kWcm for k1

and k 2 , respectively. The experiments were performed at pulse energy of 2.806 eV and

at temperatures ranging from 25 to 65 K in 10 K steps. The inset of Fig. 47 shows the PR

τ = signal intensity as a function of temperature obtained at pulse delay 12 500 fs .

3 2

1 Xh PR intensity 0 σ+σ+ 20 30 40 50 60 70 temperature [K]

T [K] 25

PR signalPR [normalized] 65 -4 -2 0 2 4 τ delay [ps] 12 − Figure 47: Traces of the diffracted 2k 2 k1 PR signal at the Xh heavy-hole exciton transition as a function τ of delay 12 recorded at excitation energy 2.808 eV and at different lattice temperatures as labeled. The co- = = −2 circular polarised pulse intensities are I1 40 , and I 2 30 kWcm , respectively. The inset shows the PR τ = signal intensity obtained at 12 500 fs as a function of temperature.

τ ≈ With decreasing temperature the signal dip at 12 0 is vanishing and the PRE signal intensity significantly decreases below 45 K. The PRE signal finally transforms to a χ(3) FWM signal below 25 K revealing the well known feature of a twice as fast signal

126 τ < τ > decay for 12 0 than for 12 0 , for a signal dominated by EID. The observed

behavior at lower temperatures is attributed to a reduced mobility of positive charges at

the ZnSe/GaAs interface by localisation. This effect hinders the redistribution of holes

and hence the stabilization of the electron grating within the QW which leads to a

decreased PR efficiency. Correspondingly, the χ ( 3 ) FWM signal becomes more

important at low temperatures.

X h T= 25 K

k k k 1 2 3

σ+ σ+ σ+

σ- σ- σ+ PR PR signal

σ+ σ- σ+ -4 -2 0 2 4 delay τ [ps] 12

+ − Figure 48: Traces of the diffracted k 3 k 2 k1 PR signal at the Xh heavy-hole exciton transition for σ +σ +σ + σ −σ −σ + σ +σ −σ + τ polarization configurations ( ) , ( ) and ( ) as a function of delay 12 recorded at

a lattice temperature of 25 K. The delay time between k1 and k 3 was kept fixed to 10 ps. The intensities = = = −2 of pulses k1 , k 2 and k3 were I1 34 , I2 24 and I 3 60 kWcm , respectively.

τ ≈ Furthermore at 12 0 an electron grating is generated within the QW by the k1 and k 2

pulse interference within the GaAs substrate that causes an exciton resonant PRE signal

in addition to the to the χ ( 3 ) signal explaining the filling of the dip at temporal pulse

overlap. Above 55 K the PRE signal intensity significantly decreases which is attributed

to a thermally activated tunneling of QW electrons back to the GaAs substrate which

reduces the lifetime of the induced QW electron grating. According to these experiments

127 there is an optimum (sample dependent) temperature balancing the hole mobility and the thermal activation of electrons to produce the highest PRE efficiency.

τ ≈ The PR nature of the signal spike at 12 0 at temperatures below 35 K is confirmed by 3-pulse experiments which were performed in a similar way as described in

= = [110]. The circular polarized pulses possess intensities I1 34 , I2 24 and

= −2 = = I3 60 kWcm (corresponding to an average pulse power of P1 20 , P2 15 and

= P3 45 W ), respectively. The delay time between k1 and k 3 was kept fixed to

τ = − ≈ 13 t3 t1 10 ps therefore excluding coherent interactions of pulse k 3 with pulses

k1 and k 2 . The delay between pulses k 2 and k1 was varied where a positive delay

τ = − > ( 12 t2 t1 )0 refers to the case where k 2 arrives last whereas a negative delay

τ < + − ( 12 )0 refers to pulse k 2 arriving first. The k 3 k 2 k1 signal response at 25 K at

+ + + − − + the energetic position of the Xh transition for configuration (σ σ σ ) , (σ σ σ ) and

σ +σ −σ + τ σ +σ +σ + ( ) are displayed in Fig. 48 as a function of delay time 12 . In ( ) and

σ −σ −σ + τ ≈ ( ) configuration we observe a pronounced PR signal peak at 12 0. As

σ +σ −σ + expected the PR signal vanishes in ( ) since k1 and k 2 have orthogonal

polarized fields that cannot produce an electron grating.

5.4.5. PCP with cw pump laser

Finally we performed two-beam experiments where carriers in the GaAs substrate are generated by cw and spatially uniform illumination at a wavelength of 637 nm (~ 1.93 eV). The focus diameter of the cw beam at the sample surface was ~ 1mm. The result of

128 -2 the experiments performed at pulse intensities I1 = I2 = 10 kWcm and at 55 K is shown in figure 49.

1.0 γ

2.0 2 P = 50 µW 0.8 [mev] 0 cw

0.6 1.6

PR Intensity 0.4 1.2 0 5 10 15 20 cw power [mW] P/P 0 cw 160 100 40 20 4

PR Intenisty [normalized] 1 -3 -2 -1 0 1 2 3 delay τ [ps] 12

P0 = 50 µW. The inset shows the exciton dephasing rate γ2 (open squares) as well as the observed PRE τ signal intensity at 12 = 500 fs (full circles) as a function of the cw laser power.

With increasing cw laser intensity (reference power P 0 = 50 µW) the exciton dephasing

γ rate 2 increases (see inset of figure 49) due to an increased electron density ne that

enhances the exciton-electron scattering rate within the QW. Above a cw power level of

7.5 mW the dephasing rate γ2 saturates which is attributed to a saturating equilibrium

τ density ne . Furthermore the signal dip at = 0 is vanishing. This is a consequence of the reduced contribution of the intensity grating at pulse overlap to the total excitation intensity. The measurements also demonstrate that the PRE efficiency can be enhanced at low cw intensity due to an increased electron density in the QW (see inset of Figure 49).

At higher cw intensities, however, the electron density modulation is significantly reduced due to the continuous and spatially uniform excitation of electrons into the QW.

129 The experiments with cw pump are consistent with the proposed mechanism of the observed PCP due to coherent exciton diffraction in the quantum well.

130 6. Summary

This thesis presents the investigations of relaxation times and coherent interactions of excitons in the Boltzmann and in the quantum kinetic regime in ZnSe single quantum wells of different well thickness. The ultra-fast method of spectrally resolved and time integrated degenerate four-wave mixing experiments with ultra-short 100 fs and 30 fs pulses has been used for these investigations. Intensity dependent FWM measurements enable to provide exciton-exciton as well as exciton-electron-hole pair scattering parameters. FWM signals at different energetic positions can be identified by quantum beat spectroscopy and polarization dependent experiments. Three-beam experiment provides more degrees freedom and enables to determine the lifetime of excitons by probing the induced grating by the third beam independently that requires no coherence to the first two beams. The thesis also explains the construction of a sub-20 fs laser

tunable in the range of 820-885 nm.

Calculations based on extended optical Bloch equations for a 10-level system have successfully reproduced and explained the characteristic features of two-beam and three-beam FWM spectra and traces obtained from 10 nm ZnSe/Zn 0.90 Mg 0.1 Se SQW

using 100 fs pulses. In particular a significant EID contribution from its polarization

dependence is observed. Intensity dependent three-pulse FWM measurements show that

γ 0 = γ 0 = -1 the zero density dephasing rate of Xh and Xl interband coherences ( 21 61 73.0 ps )

is nearly equal to that of the interexciton coherence between two Xh excitons

γ 0 = -1 γ 0 = -1 ( 32 8.0 ps ) and between Xh–Xl excitons ( 72 9.0 ps ) of opposite spin. This result indicates similar exciton-phonon and exciton-impurity interaction for all types of

131 β ≈ -1 2 coherences. In addition the Coulomb scattering for interband excitons( 21 22 s cm )

β ≈ -1 2 and that for the interexciton Xh–Xh coherence ( 32 22 s cm ) is found to be equal while

the inter-exciton Xh–Xl coherence shows a significant higher scattering rate

β ≈ -1 2 ( 72 60 s cm ) . The enhancement for the Xh–Xl coherence is attributed to an increased

Pauli repulsion in scattering events where three electrons of equal spin momentum are

involved.

The spectra obtained from the Zn 0.94 Mg 0.06 Se/ZnSe sample using ~30 fs pulses

shows the simultaneous excitation of different 1s exciton states that were identified by

comparison with calculations using the envelope function approximation. The FWM

traces of the 11h exciton transition reveal marked quantum beats and a pronounced non-

exponential decay for delay times smaller ~500 fs. The pronounced peak at pulse overlap

is attributed to a rapid destructive interference of the created polarization wave-packet

and to spin dependent quantum kinetic effects that are caused by the transient formation

of an electron-hole ( eh ) pair grating. Polarization dependent measurements support the latter assumption. Intensity dependent two-pulse experiments are performed to study the exciton-eh -pair scattering in the QW. These measurements reveal an exciton-eh -pair

β ≈ ± -1 2 scattering parameter of X −eh 90 20 s cm . The importance of the eh -pair grating generated at pulse overlap is further evidenced by intensity dependent three-beam FWM experiments. The FWM trace shows a bi-exponential decay which is attributed to a nonlinear polarization that is caused by the interaction of polarization k3 with both an exciton grating and an eh -pair grating. Model calculations assuming geminate exciton

γ −1 = formation reveal an exciton formation time of R 5 ps that is by a factor of ~4

132 shorter than the value obtained from bulk GaAs and GaAs QWs. The difference between these values is explained by an enhanced LO-phonon interaction in more polar ZnSe as well as by comparable exciton and LO-phonon energies leading to a faster relaxation of eh -pairs within the continuum.

In two- beam FWM experiment using 30 fs pulses, coherent mixed exciton-LO-

phonon polarons have been observed in a 3 nm Zn 0.9 Mg 0.10 S0.16 Se 0.84 /ZnSe single QW in which the exciton binding energy exceeds the LO-phonon energy. The formation of these superposition states is assumed to be resonantly enhanced by the vicinity of 2s exciton states. The new quasi-particles decay faster than the zero-phonon Xh transition which is

attributed to an effective disintegration of the polaron into the 1s exciton state and into

free LO-phonons as well as to inhomogeneous broadening of LO-phonon energies

leading to a fast destructive interference of LO-phonon wavepackets that couple to Xh

excitons.

A novel phase coherent PRE has been observed in ZnSe single QWs using ultrashort pulses that do not overlap in time and its spectral and thermal dependence have been investigated. The observed PCP is attributed to the formation of an electron grating formed by the interference of coherent excitons. The experimental results are reproduced by a phenomenological model that is based on optical Bloch equations for a five-level system. The high efficiency of this PCP detectable at an average power level of a few

µW makes it attractive for applications in all-optical data processing. During the exciton

coherence time the PCP enables the storage of one logic 0 bit [in the (σ −σ +σ + )

configuration] and of three different logic 1 bits [1 a in (σ −σ −σ + ) , 1 b in (σ +σ −σ + ) , and

133 1c in (σ +σ +σ + ) configurations]. The different logic bits can be distinguished by the

+ − σ +σ −σ + diffracted k 3 k 2 k1 signal intensities. The similar intense signal in the ( ) and

(σ +σ +σ + ) configurations can be further discriminated by its field polarization which is

σ − in the first but σ + in the second case. The possibility to erase the PCP by a temporal

τ ≈ overlap of the exciting pulses (i.e. 12 0) is also an interesting property for optical

switch applications.

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