AN ABSTRACT OF THE DISSERTATION OF

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2 AN ABSTRACT OF THE DISSERTATION OF Andrew D. Jameson for the degree of Doctor of Philosophy in Physics presented on January, 1. Title: Generating and Using Terahertz Radiation to Explore Carrier Dynamics of Semiconductor and Metal Nanostructures Abstract Approved: Yun-Shik Lee In this thesis, I present studies in the field of terahertz (THz) spectroscopy. These studies are divided into three areas: Development of a narrowband THz source, the study of carrier transport in metal thin films, and the exploration of coherent dynamics of quasiparticles in semiconductor nanostructures with both broadband and narrowband THz sources. The narrowband THz source makes use of type II difference frequency generation (DFG) in a nonlinear crystal to generate THz waves. By using two linearly chirped, orthogonally polarized optical pulses to drive the DFG, we were able to produce a tunable source of strong, narrowband THz radiation. The broadband source makes use of optical rectification of an ultra-short optical pulse in a nonlinear crystal to generate a single-cycle THz pulse. Linear spectroscopic measurements were taken on NiTi-alloy thin films of various thicknesses and titanium concentrations with broadband THz pulses as well as THz power transmission measurements. By applying a combination of the Drude model and

3 Fresnel thin-film coefficients, we were able to extract the DC resistivity of the NiTi-alloy thin films. Using the narrowband source of THz radiation, we explored the exciton dynamics of semiconductor quantum wells. These dynamics were made sense of by observing time-resolved transmission measurements and comparing them to theoretical calculations. By tuning the THz photon energy near exciton transition energies, we were able to observe extreme nonlinear optical transients including the onset of Rabi oscillations. Furthermore, we applied the broadband THz waves to quantum wells embedded in a microcavity, and time-resolved reflectivity measurements were taken. Many interesting nonlinear optical transients were observed, including interference effects between the modulated polariton states in the sample.

4 Copyright by Andrew D. Jameson January, 1 All Rights Reserved

5 Generating and Using Terahertz Radiation to Explore Carrier Dynamics of Semiconductor and Metal Nanostructures by Andrew D. Jameson A DISSERTATION submitted to Oregon State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy Presented January, 1 Commencement June 1

6 Doctor of Philosophy dissertation of Andrew D. Jameson presented on January, 1. APPROVED: Major Professor, representing Physics Chair of the Department of Physics Dean of the Graduate School I understand that my dissertation will become part of the permanent collection of Oregon State University libraries. My signature below authorizes release of my dissertation to any reader upon request. Andrew D. Jameson, Author

7 ACKNOWLEDGEMENTS I would like to express thanks to several people who made this work possible. First, I would like to thank my advisor, Dr. Yun-Shik Lee. His endless patience, guidance, and curiosity made the laboratory a place where I could truly flourish. To my predecessor in the lab, Jeremy Danielson, I give thanks for teaching me much and passing the torch. I also would like to acknowledge my lab partner Joe Tomaino for keeping the lab fun and whose virtuosic MatLab skills were a key component to the success of our lab work. I would also like to thank my parents, Marj and Darrol Jameson. They may have had concerns at one point about how long this work was taking, but they never flinched in their support of me, both morally and financially. Lastly, I thank the Brewstation, the best coffee house and pub in town, for providing sanctuary and family atmosphere needed to get through the stressful times.

8 TABLE OF CONTENTS Page 1 Overview of Terahertz Radiation, Sources, and Detectors Terahertz Radiation THz Sources Frequency Converting Systems Photocurrent Radiation Free Electron Sources THz Lasers THz Detectors Coherent Detectors Incoherent Detectors... 7 Electromagnetic Waves in Nonlinear Media Maxwell s Equations in Nonlinear Media Nonlinear Susceptibility Second-Order Nonlinear Susceptibility Nonlinear Susceptibility as a Tensor Phase Matching THz Generation and Detection Using a Femtosecond Laser Optical Rectification of an Ultra-short Optical Pulse Generation of Broadband THz Using Optical Rectification in ZnTe Generation of Narrowband THz Pulses Using Type II Difference Frequency Generation (DFG) DFG with Chirped Pulses Experimental Arrangement Angular Dependence of THz Output... 9

9 TABLE OF CONTENTS (Continued) Page 3.4 Terahertz Time Domain Spectroscopy With Electro-Optic Sampling Experimental Arrangement Quantitative Description of EO Sampling Distortion of the Electro-Optic Signal Intense Narrowband THz Generation via Type-II DFG Measurements and Results THz Spectroscopy of Nickel-Titanium (NiTi-alloy) Thin Films NiTi Alloy Thin Film Preparation and Composition Experimental Setup Transmission Measurements Analysis and Results Justification for the Theoretical Model The Drude Model Thin Film Fresnel Coefficients Results Transient Optical Response of Quantum Well Excitons to Intense Narrowband THz Pulses Quantum Wells Excitons Experimental Arrangement Results Experimental Results Theoretical Results and Comparison Extreme Nonlinear THz Transients in Quantum Well Microcavities... 14

10 TABLE OF CONTENTS (Continued) Page 7.1 Microcavity General Characteristics Polaritons Our Sample Experiment Experimental Setup Experimental Considerations Results Conclusions Bibliography... 19

11 LIST OF FIGURES Figure Page 1.1 Electromagnetic Spectrum Second order nonlinear effects Schematic of THz generation by a photoconductive antenna Generic four-level system for a THz laser Oscillator potentials Graphical representation of phase matching of THz radiation generated by optical rectification with optical pump Schematic of the laser system used to generate short optical pulses Optical rectification Angle-dependent THz output from optical rectification Simple schematic for generating single-cycle THz Demonstration of linear chirp Experimental setup for generating narrowband THz with Type II DFG Two orthogonal fields coincident on a [11] ZnTe crystal Experimental schematic for Electro-Optic sampling Arrangement of ZnTe crystallographic axes Index ellipse for EO sampling Filter functions for various ZnTe crystal thickness Michelson interferometer setup used to determine THz power spectra Field autocorrelation of THz pulse Comparison of Single-Cycle TDS and DFG TDS...45

12 LIST OF FIGURES (Continued) Figure Page 4.4 Demonstration of DFG tunability Emitted THz beam power as a function of central frequency Emitted THz power for both negative and positive time delays THz power vs. optical pump and Optical-to-THz conversion efficiency Sample EDX spectrum used to determine Ni and Ti concentration AFM measurements of the thickness of several NiTi alloy thin films Depiction of NiTi sample and detection schemes Raster-scan of two NiTi alloy thin films of different thickness on Si substrate NiTi alloy Relative power transmission measurement using a Si bolometer NiTi alloy time-domain data Illustration of the interface of NiTi alloy and silicon Illustration of the THz transmitted out of NiTi sample Long TDS measurements taken on NiTi alloy Summary of resistivity measurements made on NiTi alloy thin-film samples Phase diagram for NiTi alloys Simple depiction of a 1-dimensional quantum well Bulk and QW GaAs band structure Absorption spectrum of GaAs/Al.3Ga.7As QW used in our study Graphical depiction of the first two energy levels for both the infinite and finite barrier quantum well systems Experimental Arrangement for the DFG QW experiment...89

13 LIST OF FIGURES (Continued) Figure Page 6.6 Schematic of pulse compressor Internal structure of confined excitons Depiction of exciton polarization dynamics in our QW system Modulated 1-T(ω) optical transmission spectrum Several 1-T(ω) spectra Full time evolution of differential transmission Theoretical calculations of 1-T(ω) A generic resonant cavity Gaussian gain profile Polariton dispersion relation Depiction of QW micro-cavity sample Reflectivity spectrum of QW micro-cavity sample Experimental setup used with single-cycle micro-cavity QW QW micro-cavity sample mount degrees of freedom Simple white light continuum generation setup Possible quasi-particle dynamics in micro-cavity QW Microcavity QW detuning measurement Peak positions of the LEP and HEP modes vs. cavity detuning Single-Cycle THz pulse used in time-resolved study of QW micro-cavity Time-resolved differential reflectivity Time resolved reflectivity of the microcavity sample...1

14 For my wife Jennifer whose intelligence, grace, patience, and beauty are both the impetus for all I do and the standard by which I judge all I ve done.

15 1. Overview of Terahertz Radiation, Sources, and Detectors 1.1 Terahertz Radiation Terahertz (THz) radiation, also known as sub-millimeter waves or T-rays, is the portion of the electromagnetic (E-M) spectrum that falls between infrared and microwave bands (See Figure 1.1). Although there is some disagreement on the exact portion of the E-M spectrum defined as THz radiation, it is generally accepted that THz frequencies range from ~.3-3 THz..3 THz corresponds to a photon energy of about 1.4 mev, which in turn, through the Boltzmann constant, corresponds to thermal energy at a temperature of around 14 K. As a result, black body radiation in the THz can be present in all but extremely cold environments such as cryogenically controlled experiments. Although there has been much scientific interest in THz radiation since the 19 s [1], it has taken until the last few decades for the technology to emerge making production of coherent THz sources possible. During this period, THz science and technology has exploded to include a wide variety of sources, detectors, and fields of study. Figure 1.1 Electromagnetic Spectrum Because of the low energy of THz photons, they do not excite atomic transitions. This makes many dielectric materials, which are opaque to visible light, transparent to THz waves. Additionally, THz radiation is non-ionizing. Its wavelength is sufficiently short to produce detailed images of macroscopic objects. These properties have demonstrated

16 potential for the use of THz waves in security applications such as scanning for explosives and weapons [, 3], and medical applications in the non-invasive imaging of tissues [4-7]. THz radiation is also very powerful as a spectroscopic tool. THz Time Domain Spectroscopy (THz-TDS) simultaneously gathers both amplitude and phase information of the THz field. This allows direct access to fundamental material properties such as permittivity without use of complicated Kramers-Kronig calculations. THz-TDS has been applied in a variety of studies including the determination of a material s properties such as the dielectric properties of graphene [8], rotational and torsion dynamics of molecules [9,1], and identification of chemical agents such as illegal drugs [11, 1]. Another very important application of THz radiation is its use in studying carrier dynamics. Of particular interest to many research groups including our own is the use of THz spectroscopy to explore the dynamics of quasi-particles in semiconductor nanostructures. Examples of these studies include the dynamics of excitons [13-15], plasmons [16-18], and more exotic polaritons [19-1]. The relaxation times of these excitations are very fast, typically on the order of picoseconds. Even so, THz oscillations are rapid enough to explore these dynamics. 1. THz Sources Development of THz sources was driven mainly by two scientific groups: Ultrafast, time-domain spectroscopists in search of longer wavelengths, and radioastronomers who desired to work with shorter wavelengths []. From here, the development of THz sources quickly spread to many fields. Sources in use today fall into

17 3 four general categories: (i) Frequency converting systems, (ii) Photocurrent radiation, (iii) Free electron sources, and (iv) THz lasers [3] Frequency Converting Systems Frequency converting systems are one of the most commonly used table-top sources of THz radiation used for spectroscopic purposes. They use a strongly non-linear medium to convert incoming pump radiation to THz waves. Frequency conversion can occur from high optical frequencies to relatively low THz frequencies (down-conversion) or from low microwave frequencies to relatively high THz frequencies (up-conversion). Optical rectification and difference frequency generation (DFG) in a non-linear crystal are both second-order non-linear optical effects which result in frequency downconversion. Optical rectification involves a broadband, femtosecond optical pulse which creates a time-dependent polarization in a non-linear crystal. With a correctly chosen optical pump, this polarization radiates at THz frequencies, and the pulse shape is related to the incoming optical envelope. DFG involves two continuous wave (CW) optical sources, ω 1 and ω, whose frequency difference, Δω, lies in the THz range. Figure 1. depicts these processes in an oversimplified manner. Both of these methods will be discussed more rigorously in the next chapter as they were heavily relied upon in our experimentation. Frequency up conversion is achieved by creating high harmonics of microwave sources in a strongly non-linear diode.

18 4 Figure 1. Second order nonlinear effects. Simple depictions of a) Optical Rectification, and b) Difference Frequency Generation 1.. Photocurrent Radiation Another common method of producing THz radiation is through the use of a photoconductive (PC) antenna [4-6]. A schematic of a simple antenna is shown below in Figure 1.3. Figure 1.3 Schematic of THz generation by a photoconductive antenna. A bias voltage is placed across the antenna, and the gap is illuminated with a pump laser. The combination of optical excitation and the bias generates carrier motion in the gap giving rise to a time-dependent current, I(t), which emits THz radiation. As with nonlinear crystals, PC antennae can produce single-cycle THz pulses with a short optical pulse as well as CW THz radiation with two CW optical inputs.

19 Free Electron Sources Free electron sources are excellent for generating high power THz radiation [7-9]. These sources involve creation of electrons in vacuum which are manipulated by magnetic fields. The subsequent acceleration of the electrons results in the desired radiation. A Free Electron Laser (FEL) is an example of this. For broadband, short THz pulses, a population of electrons is excited with an ultra-short optical pulse. These electrons are then accelerated to relativistic speeds causing emission of radiation. CW THz can be achieved by accelerating electrons to relativistic speeds, then passing them through a magnet array causing them to oscillate in a sinusoidal pattern, which in turn radiates narrowband THz waves. These systems are widely tunable from the microwave, THz, visible, IR, and more recently to X-ray [3], and produce very high power. Their only drawback is the need for linear accelerators which requires large, expensive structures to house and run. Another free electron source is the Backward Wave Oscillator (BWO) [31]. The BWO operates on very similar principles as the FEL, but in a much more compact device. The basic principle of these devices involves an electron beam that travels in the presence of a slowly oscillating, oppositely traveling, electro-magnetic field. The EM field causes bunching and oscillation of the electrons which, in turn, emit and amplify THz radiation. Recent advances in the fields of micro-scale manufacturing have prompted predictions of miniaturization of BWO THz emitters down to the scale of handheld devices [3] THz Lasers

20 6 Directly lasing systems involve a population-inverted system whose carrier recombination directly radiates THz frequencies. Figure 1.4 below depicts this process. There are some gas and solid state systems which have carrier recombination characteristics allowing the construction of THz lasers. Recently, the most studied solid state THz source is the Quantum Cascade Laser (QCL) [33-35]. The lasing action of these devices develops from the repeated tunneling of carriers through a periodic stack of semiconductor layers. Each tunneling is associated with a THz photon, and the repetitive nature builds up a coherent THz output. QCL s provide a promising technique for miniaturizing THz sources. The only impracticality is that they are operated at cryogenic temperatures. Figure 1.4 Generic four-level system for a THz laser. 1.3 THz Detectors As mentioned above, the energy associated with THz photons is very small when compared to visible light. Because of this small photon energy, many traditional detectors are not sensitive enough for use with THz sources. However, there are several schemes which allow the detection of THz photons. Measurement methods which provide both amplitude and phase information of the THz field are referred to as

21 7 coherent detection. Systems which provide intensity information on the THz field are referred to as incoherent detectors [36] Coherent Detectors Coherent detection is a very powerful laboratory tool because, as mentioned above, it simultaneously provides amplitude and phase information of THz waves. This is important, because once the amplitude and phase information are known, the dispersion and absorption of a sample can be uniquely determined. Another key feature of coherent detection schemes is that they usually share similar aspects of the means by which the THz wave is generated. In particular, they often involve similar non-linear processes to the generation method. In addition to this, they also use a portion of the generation pump as a probe during detection. Important examples of this are electrooptic (EO) sampling in a nonlinear crystal and photoconductive switching with a PC antenna. In brief, EO sampling involves the THz field altering the index of refraction in a nonlinear detection crystal, which alters the ellipticity of a probe beam. The change in ellipticity is directly proportional to the THz field. EO sampling will be described in much more detail in chapter since it was the method used in the lab for all our time domain measurements. PC switching is similar in that the THz pump induces a current across a PC antenna which is proportional to the THz field Incoherent Detectors Because of the small photon energy, and often weak intensity of THz beams, incoherent detectors of the type used for higher energy photons have been more difficult to produce. However a group of incoherent detectors, all of which are based on thermal

22 8 effects, has emerged. The most common of this class includes bolometers, pyroelectric detectors, and Golay cells. Bolometers measure a temperature change due to a THz photon by measuring the change in resistance in a semiconductor held at cryogenic temperatures. Pyroelectric materials undergo a change in the polarization of the material due to temperature changes. Pyroelectric detectors exploit that property by measuring the small voltage change due to THz photon heating. Golay cells have a small gas chamber with a flexible diaphragm on one side. THz photons heat the gas causing a change in pressure which distorts the diaphragm. A separate reflectivity measurement off the diaphragm is taken with the change in reflectivity being proportional to the distortion. These detection methods are all sensitive to THz energy, and can be very useful down to very small THz intensities. However, since they all rely on slow thermal effects, their response times are correspondingly slow.

23 9. Electromagnetic Waves in Nonlinear Media Many methods of generating and detecting THz radiation were reviewed briefly in the previous section. Some of these methods played crucial roles in our experimentation, and will be covered in much more detail in the following chapters. These methods include generation of THz waves with optical rectification, generation of THz waves with difference frequency generation, and detection of THz waves with electro-optic sampling in a nonlinear crystal. All of these methods involve zinc telluride (ZnTe) crystals, so a later section will include calculations with the specific material properties of ZnTe. Before these calculations, it is necessary to cover some formalism of wave propagation in nonlinear optical media. The next section will present a classical wave equation/oscillator model which is sufficient to describe the light-matter interaction..1 Maxwell s Equations in Nonlinear Media The formulation for the behavior of electromagnetic waves in nonlinear media is slightly different than that for linear media, but the approach is the same. We begin with Maxwell s equations [37]: B E (.1.1) t E B J (.1.) t E (.1.3) B (.1.4)

24 For our purposes, we can consider cases in which there are no free sources of charge or current, and the materials are not magnetic, such that J =, ρ =, and B = H. 1 In addition, the relationship D E P (.1.5) will be useful, but we note that now P depends nonlinearly on the electric field, E. Specifically, P now has a nonlinear component such that D can be written as where D (1) E P (1) D (1) NL (1) NL E P P D P (.1.6) is the linear part of D. Using equation.1.6 and Maxwell s equations we can derive the wave equation in the usual way yielding (1) NL D P E (.1.7) t t Introducing the relationship (1) (1) D n ( ) E (.1.8) n n we can write the generalized wave equation as E n (1) ( n ) E n t P t NL n (.1.9) Note that the fields now bear the subscript n indicating the summation over individual frequencies. This is a necessary generalization when considering dispersive media. Now the above relationship takes the form of a driven wave equation where the nonlinear polarization is the driving term. We can further simplify equation.1.9 by making use of the following identity:

25 11 E ( E) E (.1.1) In the case of a transverse plane wave, the Laplacian of E is all that is left yielding the more familiar wave equation: E n E (1) NL (1) n n ( n ) (.1.11) t P t As will be seen in later sections, equation.1.11 can be used to qualitatively describe various nonlinear interactions. Details of this derivation can be found in [38].. Nonlinear Susceptibility In general, and as demonstrated by the simple derivation of the nonlinear wave equation above, the optical response of a material can be characterized by its polarization. We can express the polarization of a material as a power series expansion in terms of the electric field strength. For simplicity, we consider an isotropic material that is lossless and dispersionless. In this simple case, the polarization becomes: (1) () (3) 3 P( t) E( t) E ( t) E ( t)... (..1) in which the susceptibilities are scalars. The first term in equation..1 is the expression for polarization for linear media. Higher order terms represent nonlinear contributions. For even the simplest expressions of electric field, it is easy to see how various non-linear terms arise. Consider the simple example of the second order effects given by the electric field E t) E ( e ( i t e it ). (..) In this case, the second order polarization is

26 P ( t) E E E ( e e )( e e ( ) () () it it it it ) 1 (..3) which yields () () P ( t) E 1 cos( ) (.3.4) Even with this simple electric field, there are two nonlinear components that are not a function of the original frequency, ω. One is independent of frequency while the other is dependent on ω. The term independent of frequency is known as the rectification signal. Since it is independent of frequency, the rectification signal is proportional to constant field intensity, E. Optical rectification is the basis of one of our most important THz generation systems and will be discussed in more detail later. The process that produces the term proportional to ω is known as frequency doubling. The above derivation demonstrates how nonlinear processes can efficiently produce new frequencies of radiation. Another of our most important THz generation schemes depends on the related process of difference frequency generation (DFG) which will also be discussed in more detail later. Although this example is not particularly useful for real materials, it does introduce the concept of nonlinear optical effects and their proportionality to the nonlinear optical susceptibility. In reality, materials and fields are often far more complicated, and their properties must be represented by tensors. Since second-order effects are primarily involved in the THz generation and detection schemes mentioned, it is important to look more closely at the second order susceptibility..3 Second-Order Nonlinear Susceptibility A good place to begin exploring non-linear susceptibility is the Lorentz model in which the material is treated as a system of oscillators. In this case, the electrons bound

27 13 to the ion are treated as simple harmonic oscillators (SHO) driven by an electromagnetic field. However, nd order effects can only occur in noncentrosymmetric materials whose potential is not symmetric along the oscillation direction. For sufficiently weak fields (small oscillations), the SHO model is still a good approximation. As the field strength increases, the electrons are driven to larger oscillations and the SHO is no longer a sufficient approximation. One way to account for the lack of symmetry is to expand the potential in a Taylor series and include an additional term. The potential is now transformed from the normal SHO potential to the following: U 1 3 ( x) mx U( x) mx mx (.3.1) The figure below illustrates the new potential as compared with the SHO potential. Figure.1 Oscillator potentials. The symmetric SHO potential compared with an oscillator potential including an additional term. The parabolic SHO potential is a good approximation of the new potential for small values of x, but diverges as the amplitude of x becomes larger. Including this new term, the equation of motion for the system becomes: x( t) x( t) e x( t) x ( t) E( t) (.3.) t t m

28 i1t it Now consider the more generalized driving field E( t) E ( e e ) c.. 14 c Notice now that equation.3. cannot be solved exactly due to the new potential which has introduced a quadratic term in the displacement. However, if we assume αx << ω x, we can find solutions to equation (.3.) with a perturbative approach. The perturbative approach involves expanding the displacement in a power series such that x(t) = λx (1) (t) + λ x () (t) + λ 3 x (3) (t) As long as the series is convergent, this approach is valid. This means that each higher order component of x(t) must be smaller than the last (x (1) (t)>> x () (t)>> x (3) (t) ). In other words, the nonlinearity cannot be too strong, or these inequalities may not be valid. More details of this calculation can be found in [39]. In the second order of x(t), which we are primarily interested in, the solution has several separable parts that are functions of different combinations of ω 1 and ω. Of particular relevance is the portion () e * x () EE m 1 ( )( ) 1 i1 1 i1 1 ( i )( i ) (.3.3) which corresponds to optical rectification. Using the relationships P () () (, 1,, 1, ) (, 1,, 1, ) E (.3.4) and () () P (,, ) Nex () (.3.5) 1, we can show that the second order susceptibility in the case of optical rectification is: 1,

29 15 () (, 3 Ne 1 1, ) m ( 1, ) 1,, 41, (.3.6) All the other portions of the second order susceptibility mentioned above can be found in a similar manner, as well as higher order susceptibilities if desired..4 Nonlinear Susceptibility as a Tensor Before progressing to the specific detection and generation methods used in my research, we need a more rigorous definition of the susceptibility as it applies to materials which are not isotropic. In general, the second order nonlinear polarization can be written as: P ( i n () ) D (,, ) E ( ) E ( ) (.4.1) m jk mn ijk where D is a degeneracy factor that comes from summing over the frequency components. Equation.4.1 can be extended to arbitrary order by simply adding more frequency components and more field components. Notice that in this case the tensor can potentially have 7 independent components. We can simplify this using what is known as contracted notation. First we introduce the definition: n m n m j n k m () ijk dijk 1 () ijk (.4.) We now apply symmetry conditions, known as Kleinman s symmetry conditions, which greatly reduce the complexity of the tensor [4]. These conditions require d ijk to be symmetric in its last two indices such that d ijk = d il. The simplified notation is as follows:

30 16 jk : l : ,3 4 13,31 5 1,1 6 (.4.3) Applying.4.3 reduces the number of independent components to 18. Applying Kleinman permutation symmetry [4], it can be shown that the total remaining independent components in contracted notation is only 1 such that: d11 d1 d13 d14 d15 d16 d il d16 d d3 d4 d14 d1 (.4.4) d15 d4 d33 d3 d13 d14 Equation.4.4 can be used in matrix calculation of values of the polarization. As will be shown, many crystals have additional symmetry that further reduces the number of independent components of the susceptibility tensor..5 Phase Matching As we have discussed the background of electromagnetic radiation propagating in nonlinear media and some of the effects that arise as a result, it is important to discuss phase matching. Consider two EM fields E ~ ~ r) A ( )exp( ik r), E ( r) A ( )exp( ik ) (.5.1) 1( r where A i are complex amplitudes which are functions of different frequencies ω 1 and ω. As the two waves propagate through a material, perfect phase matching is defined by the simple relationship k k 1 k. (.5.) Depending on the situation, phase matching may be referring to different processes, but this relationship is always valid. For example, the situation may be a sum frequency generation (SFG). Using the fields above, the output of such a process would be

31 E 17 ~ r) A ( )exp( ik ) (.5.3) 3( 3 1 r where the efficiency of the output depends on the value of k. Another situation, which will arise later when discussing optical rectification in ZnTe, is the phase matching of a pump field and the field it generates. In this case, the optical pulse and the THz pulse propagate together. If the phase velocity of the THz pulse matches the group velocity of the optical pulse, coherent amplification of the THz radiation will occur. If these velocities are not matched, then destructive interference will eventually dominate the process, and the output will be weak or nonexistent. Phase matching and phase mismatch is graphically depicted below in Fig... Fig.. Graphical representation of phase matching of THz radiation generated by optical rectification with optical pump. (a) Optical group velocity is equal to THz wave phase velocity resulting in amplification of the THz radiation. (b) Velocity mismatch between THz wave and optical pump results in interference and weak signal.

32 18 In principle, it is difficult to perfectly phase match multiple traveling waves inside a material. Phase matching is especially difficult for short pulses since they contain a band of frequency components which will travel at different velocities due to dispersion. Additionally, for nonlinear processes, the output is often of a much different frequency, and most materials do not have a flat frequency response over many orders of magnitude. If the nonlinear medium is too thick, these destructive processes will dominate. If the medium is too thin, coherent amplification will not build up, and the output will be weak. In practice the thickness of the material must be balanced to maximize the output while minimizing parasitic effects. The parameter by which this is characterized is called the walk-off length. In general, the walk-off length can be expressed as [41]: l w n THz c p n opt (.5.4) However, we must keep in mind that the optical index of refraction that we refer to is the group index. So, in general, the task of finding a good nonlinear material comes to finding one with optical properties that give a large walk-off length.

33 19 3. THz Generation and Detection Using a Femtosecond Laser With the background of chapter, we can now explore in more detail a few specific nonlinear optical processes that are central to this research. These include the generation of THz waves through optical rectification of an ultra-short optical pulse, the generation of THz waves through type II difference frequency generation (DFG) of two linearly chirped, orthogonally polarized pulses, and the detection of THz waves using electro-optic sampling. 3.1 Optical Rectification of an Ultra-short Optical Pulse Optical rectification of an ultra-short optical pulse to generate THz radiation is the work horse of the lab. This method generates a relatively broadband, strong, singlecycle THz pulse that was useful for everything from characterizing materials with linear transmission measurements to exciting nonlinear carrier effects. To demonstrate how THz waves are generated, it is important to know the properties of our laser system. The simple schematic below shows the important parts of the system: Figure 3.1 Schematic of the laser system used to generate short optical pulses.

34 As Fig. 3.1 shows, the output of our laser system is a high-power, ultra-short, optical pulse. Previously, we only considered monochromatic plane waves as the incident radiation. Now we treat the incident field as a transform-limited, Gaussian pulse which approximates the output of our laser system very well. In the time domain, this pulse has the following form: E ( t) E cos( )exp t t (3.1.1) In Chapter, by deriving the wave equation (equation.1.9) for nonlinear media, P NL n we saw that the nonlinear polarization, acts as a source term for the t propagating wave. Using equation.3.4 from the previous chapter, we can rewrite the wave equation in the following form: E n (1) (1) E E n () opt ( n ) (3.1.) t t Here, we have assumed μ = 1 since the materials we will discuss have negligible magnetic properties. Qualitatively, we can see an immediate relationship between the polarization and radiation it produces. The radiated field (E n = E THz in our case) is proportional to the second time derivative of the intensity of the driving optical field, E opt. Taking the second derivative of equation 3.1.1, we have: E opt t 16t 4t t E THz E exp exp (3.1.3) 4 t t

35 Based on equation 3.1.3, the figure below illustrates the shape of the THz waveform we can expect as a result. 1 Figure 3. Optical rectification. Optical rectification produces a THz pulse whose waveform is proportional to the second time derivative of the optical pulse envelope. 3. Generation of Broadband THz Using Optical Rectification in ZnTe In general, the THz output due to optical rectification depends on the polarization of the driving optical pulse with respect to the crystallographic axes. It is therefore necessary to use the tensor formalism for susceptibility introduced in Chapter to characterize the THz radiation. In general, the nd order tensor polarization for optical rectification can be expressed as [41]: E x Ey Px d11 d1 d13 d14 d15 d16 Ez P y d1 d d3 d4 d5 d6 (3..1) E yez Pz d31 d3 d33 d34 d35 d36 ExEz ExEy

36 As ZnTe used as a source in nearly all our experiments, it is worth discussing in more detail. ZnTe is cubic crystal of the class 43m. This crystal class has only three non-vanishing matrix elements in its contracted susceptibility tensor which are all equal such that d 14 = d 5 = d 36 [41]: d14 d il d14 (3..) d14 In spherical coordinates, an arbitrarily polarized field is given by: sin cos E E ˆ E sin sin (3..3) cos Using 3..1, 3.., and 3..3 the polarization becomes: Px Py E Pz 4 d 14 sin cos sin sin d14 cos d 14 sin sin cos d14 sin cos cos sin sin cos sin cos E sin cos cos sin sin cos (3..4) To observe nd order nonlinear effects we wish to maximize the THz intensity. As the THz radiation is parallel to the polarization, we can use the expression for

37 3 polarization intensity to find the optimal alignment of the optical pump. The intensity of the polarization is: 4 P d E sin 4cos sin sin (3..5) 14 Equation 3..5 is maximized for sin (φ) = 1, or φ = (n+1)π/4 for n =,1, which yields: 4 I THz (, ) P d E sin 4cos sin (3..6) 14 which can be written as 3 MAX I THz (, ) P I THz sin 4 3sin (3..7) 4 Equation 3..7 has a maximum at sin 1 3. In terms of a ZnTe crystal cut in the common fashion shown in the Figure 3.3 below, this tells us we can maximize the output of optical rectification by aligning the optical pump polarization along the [ 111] axis, or at about θ = from the [1] axis.

38 4 Fig. 3.3 Angle-dependent THz output from optical rectification. (a) A [11] cut ZnTe crystal. The white arrow represents the propagation direction of the optical pump, and θ is the polarization angle with respect to the [1] axis. (b) Output THz intensity as function of θ. The figure below illustrates the simple experimental setup required to generate a singlecycle THz pulse via optical rectification. Also shown are a comparison of the theoretical angle-dependent intensity with experimental data, and an example of typical single cycle output obtained by electro-optic sampling which will be discussed later in this chapter.

39 5 Fig. 3.4 (a) Simple schematic for generating single-cycle THz. (b) Comparison of theoretical and experimental angle-dependent THz intensity. (c) Sample THz pulse generated using optical rectification. Fig 3.4(b) shows that the measured THz intensity deviates slightly from the theoretical value. This deviation from theory is due to unaccounted for higher-order effects and inhomogeneity in the generation crystal. As can be seen in Fig 3.4 (c), the shape of the THz pulse generated via optical rectification matches fairly well with the theoretical shape in Fig. 3. with some deviations. These include a slightly narrower and deeper oscillation after t =, and some subsequent ringing oscillations that follow. The inhomogeneity of the pulse just after t = is due to phase mismatch discussed in section.5, while the subsequent high frequency ringing is caused by the increase in index of refraction as the THz frequency increases [4]. Further oscillations later in time are often due to absorption and reemission of the THz after it leaves the crystal. These oscillations are especially noticeable when water vapor is present as it strongly absorbs THz radiation. The ringing is commonly countered by purging the experimental area with dry nitrogen or performing the experiment in an evacuated chamber.

40 3.3 Generation of Narrowband THz Pulses Using Type II Difference Frequency Generation (DFG) 6 As mentioned in section.3, when a sufficiently intense field of the form E( t) E(exp( i1t ) exp( it) c. c) impinges on a nonlinear crystal, several secondorder nonlinear processes can occur including difference frequency generation and optical rectification. As an example, the frequency independent case of optical rectification was discussed in more detail. Now we look at the process of DFG which involves frequency dependence proportional to exp t i 1. Simply put, DFG involves the input of two sources, ω 1 and ω, which generate an output of frequency ω 3 = ω 1 - ω. Producing THz waves with this method has been around for over fifty years. Zernike et al used a Nd:Glass laser and a quartz crystal to produce an output at 1 cm -1 (1 μm or 3 THz) in 1965 [43]. More recently, similar tunable THz sources have been created which use DFG with a variety of sources and materials [44-47] DFG with Chirped Pulses In order to generate narrowband THz pulses, our group developed a novel, tunable, table-top system. As will be shown, the development of our technique yielded excellent results [48]. This method involves using two linearly chirped, ultra-short optical pulses. To create a chirped pulse, one simply needs to introduce some group velocity dispersion (GVD) into the pulse. Adding or compensating for GVD is most commonly achieved by the use of prisms or diffraction gratings. A set of either of these devices can introduce a linear chirp by causing a delay between the high frequency

41 7 components of the pulse and the low frequency components. Equation gives us the field for the transform limited case. To express the chirped pulse, we need to add a linear phase term such that t E( t) E exp exp i bt t (3.3.1) The figure below demonstrates the results of a linear chirp on a transform-limited pulse. Fig. 3.5 Demonstration of linear chirp. (a) Transform limited optical pulse. (b) The same pulse with a linear chirp added in which high frequency components are moved to the right, and low frequency to the left. By looking at figure 3.5(b), we can see that the frequency components have been spread out in time. The instantaneous angular frequency of this pulse is defined as t bt bt d d ins ( t) ins (3.3.) dt dt demonstrating that the frequency of the pulse increases with time if b is positive. By combining two such chirped pulses with a time delay, τ, between them, it is possible to

42 overlap different frequency components in time, enabling DFG. With this configuration, the instantaneous frequency difference between the two pulses is: 8 1 b f ins bt bt (3.3.3) Equation suggests that with a specific chirp parameter, b, and time delay, τ, Δf ins will be constant in time, and thus the resulting DFG polarization will oscillate at this same frequency throughout the duration of the overlap Experimental Arrangement Our experimental arrangement is as follows. By not fully recompressing the output of our regenerative amplifier, we generate a chirped pulse. This pulse is split with a 5-5 beam splitter (BS), and the polarization of one pulse is rotated 9 by a double pass through a λ/4 wave plate. A delay is introduced between the pulses, and they copropagate through a 1 mm ZnTe crystal where Type II DFG generates a narrowband THz pulse. A depiction of the experimental setup is shown below in Figure 3.6. There are noteworthy advantages to using this configuration. The first is that if a typical Michelson interferometric setup were used, the maximum beam intensity that can be output is 5% of the input due to reflective losses. By using orthogonally polarized light and a thin film polarizer (TFP), transmitted power was maximized. The other advantage to this set up is that it minimizes parasitic effects, such as two-photon absorption, that decrease the THz conversion efficiency [48].

43 9 Fig. 3.6 Experimental setup for generating narrowband THz with Type II DFG Angular Dependence of THz Output As in the previous section, we can use the tensor formalism to characterize the angular dependence of the DFG output. The general form of the fields can be derived from Fig. 3.7 below. For this general arrangement, the polarization vectors for the fields E 1 and E can be written in terms of Cartesian coordinates as: E E x y cos z sin x ysin z cos (3.3.4) Here we have neglected to include the time dependence and amplitude for simplicity. Making use of equation and equation.4.1 in matrix form, we can again write an

44 3 expression for the polarization which is somewhat more complicated than that for optical rectification since we are now pumping with two independent fields in the configuration shown below. Fig. 3.7 Two orthogonal fields coincident on a [11] ZnTe crystal. Again, making use of the d-matrix for ZnTe, the expression for the polarization is: ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( x y y x x z z x y z z y z z y y x x z y x E E E E E E E E E E E E E E E E E E d d d P P P (3.3.5) Using equations and performing a bit of algebra we can write equation as E E

45 31 Px Py E1E Pz * exp( i3t) d 14 d 14 cos sin cos sin cos sin cos d14 cos sin (3.3.6) where we have included the time dependence of the field in which ω 3 = ω 1 ω. Equation can be simplified to Px Py Pz * d14e1e exp( i3t) cos cos sin (3.3.7) As before with optical rectification, the THz output is proportional to P, which gives us an angular dependence for the DFG THz wave defined by 1 E P 3cos THz (3.3.8) Equation tells us two things. First, we notice that the amplitude of E THz is maximized when =. In other words, E 1 and E must be aligned along the 1 1 and 1 axes respectively to maximize the THz output. Second, we see that we should expect a symmetric sinusoidal variation of the field amplitude with unlike optical rectification. As the experimental output of this setup constituted my first published work, the results of this method are discussed in detail in Chapter Terahertz Time Domain Spectroscopy With Electro-Optic Sampling

46 3 One very powerful measurement tool used daily in our studies was terahertz timedomain spectroscopy (THz-TDS). As we will see, THz-TDS allows us to map out the THz field amplitude in time. The reason this tool is so potent is that it simultaneously delivers both the amplitude and phase information of the THz field. The availability of the phase and amplitude information grants direct access to electronic properties of materials, such as permittivity, without the cumbersome Kramers-Kronig calculations necessary from traditional spectroscopy [49-5]. The phase and amplitude of the electric field is mapped out by use of the linear electro-optic (EO) effect, also called the Pockels Effect. The EO effect will be quantitatively described in the next section. The EO effect, in essence, is a change in index of refraction of a material proportional to a constant or low-frequency electric field. In our case, the THz field induces a birefringence in an EO crystal (in our case ZnTe). A low power, linearly polarized optical probe is sent through the EO crystal simultaneously. Due to the birefringence, the probe polarization will be rotated by some amount directly proportional to the THz field. By changing the delay between the optical probe and THz pump, one can sweep the probe across the entire THz pulse essentially mapping out the field amplitude Experimental Arrangement Figure 3.8 below demonstrates a typical setup to perform EO sampling.

47 33 Fig. 3.8 Experimental schematic for Electro-Optic sampling. This figure demonstrates the polarization of the optical pulse after each component (The dotted circle demonstrates the slight deviation from circular polarization). In the absence of the THz field the linearly polarized probe pulse is not rotated. A λ/4 plate transforms the pulse to circular polarization which is split into orthogonal components by means of a Wollaston prism. These are measured by a balanced photodiode which will register zero current in this case since circular polarization has equal orthogonal components. In the presence of the THz field, the output of the EO crystal will be slightly elliptical. This slight ellipticity translates to a slightly elliptical input into the Wollaston prism, which will register as a small current in the photodiode since one signal is now slightly larger than the other. The difference is proportional to the THz field amplitude at that particular delay position. As mentioned above, one can change the placement of the probe pulse with respect to the THz pulse logging these differences at each delay position effectively mapping the THz waveform as a function of time Quantitative Description of EO Sampling

48 34 Developing a more quantitative approach to EO sampling not only helps to understand the underlying physical mechanisms, but allows us to optimize the experimental arrangement. Unlike the previous sections, the linear EO effect has traditionally been described in a different manner that will be followed here [53]. In general, the relationship between E and D for an anisotropic material can be represented by the following equation: D D D x y z xx yx zx xy yy zy xz E x yz E y zz E z (3.4.1) For a lossless material, the dielectric tensor is real and symmetric, which means that it can be expressed in a diagonal form by means of an orthogonal transformation [53]. In other words, there is a rotation of the current axes possible that will diagonalize the permeability tensor. The new coordinate system is known as the principle-axis system, and yields the following expression in terms of the new axes: D D D X Y Z XX YY E X EY ZZ EZ, x X y Y z Z (3.4.) Using the above expression and considering the energy densityu 1 D E, we can derive an expression for the surfaces of constant energy in terms of the new directions X, Y, and Z:

49 35 X n XX Y n YY Z n ZZ 1 (3.4.3) where we have used X D X 1, Y D Y 1, Z D Z, and ij n ij (3.4.4) Equation is known as the index ellipsoid since it describes the shape of an ellipse. It can be used to relate the specific axes of the material to their respective indices of refraction for a given propagation direction. Note that for a set of coordinates which are not the principal axes, this expression will be complicated by cross terms since the permeability tensor is not diagonal. The most general form of the index ellipsoid is x y z xy yz xz 1 (3.4.5) xx yy zz xy yz xz 1 where we have defined the impermeability tensor as. ij ij The next step is to examine what happens to the index ellipsoid when an electric field is applied to the material. The arrangement of the field and crystal are as shown below:

50 36 Y 1 Z 1 1 E THz φ X 11 Fig. 3.9 Arrangement of ZnTe crystallographic axes. The ZnTe crystal is cut along the [11] axis to maximize nonlinear effects. The THz is aligned as shown and propagating along the [ 1 1] axis. The detailed procedure by Casalbuoni, et al in reference [4], shows how this formalism applies specifically to our case of a THz field in a ZnTe detection crystal. Following their process, we expand the impermeability tensor in the applied field. We assume that this series converges and retain the first two terms () ij ij ij r ijk Ek... (3.4.6) k which will be greatly simplified by the fact that ZnTe has only three non-zero terms. Expanding equation in matrix form, we find that the permeability is no longer diagonal. As a result, we must find the normalized eigenvectors which point along the direction of the principal axes. These are given by Casalbuoni et al as

51 U U U sin 1 3cos sin 1 3cos 1 1 cos 1 3cos sin 1 1 cos 1 3cos sin 37 (3.4.7) derived from the impermeability eigenvalues: 1, 1 n 1 3 n r r E E THz THz sin sin 1 3cos (3.4.8) Note that the first order parts of the eigenvalues are all equal to 1/n since ZnTe is isotropic in the linear regime. From the principal axes, we immediately note the U 3 points normal to the [11] plane (parallel to the propagation direction of the E THz ) and thus U 1 and U are the axes that define the field-induced birefringence. U 1 lies in the [11] plane, but at some angle, ψ, to the [ 11] axis defined by sin cos (3.4.9) 1 3cos A schematic of the resulting axes along with the THz field is shown below.

52 38 Y [1] U 1 U n 1 E THz n X [11] Fig. 3.1 Index ellipse for EO sampling. The index ellipse for a [11] cut ZnTe crystal pumped by a THz field propagating along the [ 1 1] axis. The resulting birefringence is shown exaggerated along the new principal axes U 1 and U. If we return to equations and recall that ij 1 n ij, we can obtain expressions for the index of refraction along U 1 and U. Since η ij converges, we can use r41 ETHz / 1 n to obtain the following expressions for n 1 and n 3 nr41e n 1, n THz sin 1 3cos (3.4.1) 4 For a crystal of thickness d, an optical probe pulse will accumulate a phase difference between its two orthogonal polarization vectors. The phase difference, designated Γ, due to equation is 3 d dnr41ethz n1 n 1 3cos (3.4.11) c c

53 Equation shows that the phase shift is indeed directly proportional to E THz as stated earlier. Clearly Γ is maximized at φ =, which corresponds to a maximally observable effect if E THz is polarized along the [ 11] axis. In this maximized case, U 1 would be at an angle ψ = Distortion of the Electro-Optic Signal In practice, the EO signal mapped out by the balanced photodiode can be quite distorted with respect to the actual THz waveform entering the EO crystal. The degree of distortion depends on several factors including (i) the bandwidth of the optical probe in the form of the autocorrelation of the probe spectrum, (ii) the dispersion of the nd () order susceptibility, ( ), and (iii) the phase mismatch between the THz and optical probe [4]. The EO signal can be expressed as: S( ) ATHz ( ) f ( )exp( i ) d (3.4.1) where f () is given by () exp( ik (, ) d 1 f ( ) COpt ( ) ( ;, ) (3.4.13) ik (, ) In this formulation [54, 55], S(τ), is the signal after the EO crystal, A THz, is the THz spectrum entering the crystal, and f(ω) is the filter function of the EO crystal. The filter function consists of the three components listed above (C opt is the spectrum of the autocorrelation of the optical probe, χ () is the dispersive nd order susceptibility, and the final expression in brackets is the phase mismatch characterized by Δk +. d is crystal

54 thickness). Taking into account the THz absorption in the EO crystal, the following figure shows the filter function for several crystal thicknesses. 4 F mm Frequency (THz) Fig Filter functions for various ZnTe crystal thickness. Reproduced with permission from [56]. The plot brings to light many practical considerations for EO sampling. We notice that the filter function is smoothest over the largest range of THz frequencies for thin crystals because of the minimization of phase mismatch. Above a frequency of about 4 THz however, the sensitivity fails due to the absorption wing of the transverse optical (TO) phonon absorption at ~5.3 THz [56]. As the crystal thickness increases, we notice that the bandwidth of the sensitivity decreases due to increased phase mismatch, and that the acoustic phonon absorptions at ~1.6 and ~3.7 THz further distort the filter function [54]. These distorted filter functions can cause distorted waveforms. From equation , we

55 41 also see that the EO signal will be proportional to the crystal thickness, d. Depending on the goals of experimentation, an appropriate thickness must be chosen for a detection crystal to optimize for signal strength, bandwidth, signal fidelity, or a combination thereof. Of course, experimentally one would like to have the EO output be exactly characteristic of the input, especially when the experiment involves passing THz waves through a sample and measuring the results via THz-TDs. It has been shown [55] that even when distortion is present, sample properties can be extracted by use of a reference. Now that we have laid down a fairly comprehensive groundwork on the background and relevance of THz radiation and the main tools used in our lab, we can explore, in detail, the results of experiments I have performed.

56 4 4. Intense Narrowband THz Generation via Type-II DFG As this subject is not only one of the fundamental tools used in my experimentation, but also my first published work, the experimental results have been given their own section for discussion here in chapter 4. Both broadband and narrowband THz sources are important tools in our laboratory. Depending on the type of experiment, it is beneficial to have a narrowband source for exciting specific effects whereas a broadband source may excite a number of effects simultaneously. A few pertinent examples of these experiments include studying transitions among impurity states in semiconductors[57], intra-band transitions of excitons in semiconductor nanostructures [58,59], and many-body interactions of strongly correlated carriers [6]. However as covered in Chapter 1, until recently, there did not exist many THz devices (sources, detectors, optics, etc). As such, there were very few compact, tunable sources of narrowband THz radiation available. FEL s are intense and tunable, yet their accessibility is limited due to their size. Molecular gas lasers are compact sources of intense THz radiation [61], yet they lack tunability. There were a few sources available that offered both the desired compactness, and tunability. These were mixing of chirped optical pulses in a PC antenna [6], optical rectification of shaped pulses [63], and the optical rectification in quasi-phase-matching nonlinear crystals [64]. These all provided the added beneficial feature of easily adapting to phase-locked, timeresolved studies. However, at the time, these available sources lacked the ability to generate the necessary intensities we needed to induce nonlinear effects. This lack of

57 43 intensity led us to develop our own table-top, tunable source of narrowband THz which is outlined in Chapter 3 based on the chirped-pulse/pc antenna arrangement. Here we discuss the capabilities of this powerful system. 4.1 Measurements and Results The experimental arrangement for this method is shown in Figure 3.5. Some advantages of this setup are listed in section 3.3. which includes minimization of parasitic effects, and maximization of optical pump throughput. Additionally, by replacing the PC antenna as the source of THz waves with a ZnTe crystal, we avoid some drawbacks of the antenna. As our pump source pulse energy is ~1 mj, PC antennas are either incapable of sustaining this high peak power, or will become inefficient due to saturation effects. Additionally, due to finite carrier response times of the substrate, PC antennas suffer a large decrease in efficiency above ~1 THz [48]. The first measurements were done to determine the spectral output of the DFG using a Michelson interferometer. The setup of the interferometer in conjunction with a bolometer and a typical output are shown in the figures below:

58 44 Fig. 4.1 Michelson interferometer setup used to determine THz power spectra. An example of a typical field autocorrelation and its corresponding power spectrum obtained through FFT is shown below in Figure 4.. Fig. 4. Field autocorrelation of THz pulse. Data generated by the Michelson interferometer in Figure 4.1 at pulse length 4.11 ps, and delay of 1.9 ps. Inset shows the corresponding power spectrum.

59 45 Before we move on to other characterizations of this THz source, it is important to be clear on the difference between τ and τ p in Figure 4. so as not to confuse them. τ is the delay introduced between the two orthogonal pulses, while τ p is the pulse duration which is controlled by a pair of diffraction gratings present in the regenerative amplifier. Figure 4. was generated with a pulse duration of τ p = 4.11 ps, and a delay of τ = 1.9 ps. The spectrum (and its waveform) can also be generated via THz-TDS. Figure 4.3 below shows a comparison of a single-cycle pulse with its broadband spectrum and a manycycled pulse generated with the DFG and its corresponding narrowband spectrum. Fig. 4.3 Comparison of Single-Cycle TDS and DFG TDS. (a) Comparison of waveforms in the time domain, and (b) comparison of the spectra in the frequency domain. White corresponds to the single-cycle THz pulse in both cases. From Figure 4.3(a) we note that the pulse duration of the single-cycle pulse is approximately ps, while the multi-cycle pulse lasts around 5 ps. Additionally the

60 full-width half-max (FWHM) of single-cycle spectrum is approximately 1.5- THz while the FWHM of this particular DFG spectrum is only ~.4 THz. 46 As was mentioned in Chapter 3, the difference in frequencies of the two chirped pulses is linearly dependent on the delay between them. By changing the delay, we tune the output frequency of the setup. Tunability is clearly demonstrated in Figure 4.4 below. Fig. 4.4 Demonstration of DFG tunability. The output of the DFG can be tuned by varying the delay between the two chirped pulses. The inset shows the center frequency as a function of delay. Figures generated using the Michelson interferometer setup.

61 47 Figure 4.4 clearly demonstrates how powerful a tool this THz generation method can be. Since the setup is capable of fine and continuous frequency tuning, it is possible to excite specific resonances in physical systems not possible with a broadband pulse. Exciting specific resonances is especially important when the need arises to separate the many simultaneous effects induced by a broadband pulse (such as a sample whose behavior is unknown). The inset to Figure 4.4 shows that the center frequency of the THz varies linearly with pulse delay as predicted by equation The solid line represents a linear best fit, and hence yields a chirp parameter of b = 3.85 ps. Next, we examine the power dependence as a function of THz center frequency by using several fixed pulse durations, and scanning along several values of delay. Figure 4.5 shows the measured power spectra for several pulse durations. Fig. 4.5 Emitted THz beam power as a function of central frequency. Data for time delays of τ p = 1.6,.3,.78, 3.35, and 4.61 ps.

62 48 Figure 4.5 demonstrates several features of this THz source. First, the peak power decreases as the pulse duration increases. Additionally, for every pulse but the longest, the peak power occurs around 1 THz, whereas for τ p = 4.61 ps, the peak power is centered near 1.5 THz. In general, the shape of the spectrum is limited by two factors. The first is that Type II DFG is a second-order nonlinear process. Thus, the power is a quadratic function of the frequency, and the spectrum to falls off sharply for low frequencies because of this. Secondly, to increase the frequency, we must increase the delay time between the two pulses. As the delay increases, the two pulses overlap less and less in time, and the DFG conversion quickly gets weaker. As a result, the spectrum falls off rapidly for high frequency components as well. The only time this does not occur is for τ p = 1.6 ps. We clearly see that the spectrum does not go to zero at both low and high frequencies. As the shortest pulse, this also corresponds to the least chirped pulse, or conversely the most compressed. These low and high frequency power components can be explained by additional THz power being generated due to optical rectification. As we see from equation 3..7, the beam aligned to the [ 1] axis generates no THz intensity by these means, thus optical rectification is due only to the orthogonal beam aligned along the [ 11] axis. This alignment is not the optimal axis for THz generation from optical rectification, so this THz is substantially weaker than the DFG radiation. However, as Figure 4.3 demonstrates, the frequency response for optical rectification is much broader which accounts for the nonzero low and high frequency components.

63 49 If we expand these measurements to include both positive and negative time delays, we see some additional interesting structure to the power spectra. Figure 4.6 again shows several fixed pulse durations now with longer scans over positive and negative time delays. Fig. 4.6 Emitted THz power for both negative and positive time delays. The inset shows optical rectification power for positive and negative time delays. As outlined in section 3.3, E 1 corresponds to the polarization along the [ 11] axis, while E corresponds to the polarization along the [ 1] axis. Negative time delay is associated with the case in which E 1 precedes E. Using Figure 4.6, we notice a definite imbalance in the THz power depending on the pump pulse order which becomes stronger as the pulse duration becomes shorter. The weaker DFG THz generation in the negative time

64 5 delays is due to anisotropy in the nonlinear absorption processes. To understand this asymmetry, we look at the inset of the figure which shows the THz generation for two orthogonal, fully compressed pulses. For fully compressed pulses, the generated THz power is due entirely to optical rectification. For positive time delays, E precedes E 1. Recall again that E does not generate THz waves due to optical rectification. However, nonlinear absorptions of the optical beam leads to free carriers which heavily absorb both E 1 and the THz it may generate leading to a weak THz signal for positive time delays. In the opposite case, E 1 generates THz power more efficiently through optical rectification; hence there is less nonlinear absorption to leech away the generated THz photons. For DFG, the effect is different with the result being opposite. In this case, THz photons are only generated (excepting the relatively weak rectified signal for the shortest pulse durations) when the two pulses overlap. It has been shown that some zincblende crystals have a stronger multi-photon absorption along the [ 11] axis [65]. Since there is no rectified THz generation to share the pump power as in the fully compressed case, E 1 will eventually have larger parasitic nonlinear absorption when it precedes E. When E 1 arrives first, the free carriers generated by E 1 will tend to absorb some of E and the THz generated by the overlapping pulses resulting in a lower power for negative time delays instead. The asymmetry tends to weaken as the pulse duration is increased. Finally, we look at the THz power vs. Optical pump power and the Optical-to- THz conversion efficiency for several pulse durations.

65 51 Fig. 4.7 THz power vs. optical pump and Optical-to-THz conversion efficiency. For pulse durations τ p = 1.6,.13,.78, 3.35, and 4.61 ps: (a) THz power vs. Optical pump power, and (b) THz conversion efficiency vs. Optical pump power. We see that in the range of powers we used, the emitted power reached a saturated value. Below a certain value of pump, with an average intensity of 3.8 W/cm, the terahertz power varies quadratically with the input power, as expected for a second-order process. Above this pump level it varies almost linearly with input power. The saturation of the conversion efficiency can be attributed to the parasitic nonlinear effects in the ZnTe crystal. In conclusion, we have demonstrated and characterized a source of intense, narrowband THz pulses. It has several distinct advantages over other sources of narrowband THz including, economy of space, maximization of optical throughput, and minimization of parasitic nonlinear optical effects. Additionally, it is easily incorporated

66 5 into an ultrafast spectroscopy setup and ideal for ultrafast time resolved measurements as the pulse durations are still on the order of picoseconds. In Chapter 6, we will see one example of the power and usefulness of this setup as it is applied to semiconductor nanostructures. This work was published in the Journal of Applied Physics in 8 [48].

67 53 5. THz Spectroscopy of Nickel-Titanium (NiTi-alloy) Thin Films NiTi-alloy, sometimes referred to as Nitinol when the proportions of each metal are near 5%, is an intriguing material. It is what is known as a shape memory alloy (SMA). SMA s have multiple stable crystalline structures which depend on the temperature of the alloy. At some critical temperature, T c, the alloy undergoes a transition from the low temperature martensitic state to the high temperature austentitic state [66]. What this means is that when the crystal is annealed above T c in shape 1, it can be cooled below T c, and deformed into shape. Then, when the material is heated above T c, it will spontaneously return to shape 1. Shape memory is one of many interesting qualities NiTi alloys possess including superelasticity and electroplasticity [67-7]. As a result of these interesting characteristics, NiTi alloys have been exploited for a variety of applications including medical implants and microelectromechanical systems (MEMS) [71-74], and the micromechanical properties of this alloy are well studied [56, 75-78]. If very small devices in the nanometer to micron range are being developed, especially those which may be electrically heated to trigger shape memory effects, it is important to accurately characterize the electrical properties of NiTi alloys. Accurate characterization of the electrical properties is especially important for thin metal films which have been shown in many cases to deviate from bulk measurements of electronic properties [79-88]. Despite this, the electrical properties of NiTi alloy thin films remains unstudied. Electrical conductivity (or, inversely, the resistivity) has been reported as a good property

68 54 with which to characterize the phase transitions and micromechanical properties of NiTi alloys [89]. We used THz imaging and THz-TDS to measure the resistivity of NiTi alloy thin films of varying thickness (~3-1 nm), and varying concentrations of Ti (-1%). As is illustrated later in figure 5.1, this yielded two sets of comparable resistivity data. One was derived from the time-averaged power transmission (bolometer, which will be referred to as power transmission ), and one from the field amplitude transmission (EO sampling). We also performed four-point-probe measurements on the samples for comparison. Although four-point-probe measurements are typically accurate, THz-TDS has also been shown to be an effective method for characterizing the properties of metal thin films [87, 89, 9-9], and has the added advantage of being non-intrusive when compared to the four-point-probe method which effectively destroys the sample. 5.1 NiTi Alloy Thin Film Preparation and Composition This section briefly outlines how the samples were created, and how the composition (Ti% and film thickness, d) of the samples were characterized in collaboration with the Minot group from the Oregon State University Physics department [93], and the Koretsky group from the Oregon State University Chemical Engineering department [94]. The samples were prepared using a fairly standard lithographic method. After application of a photoresist layer, a shadow mask was applied to the sample in a SUSS MJB 3 contact aligner, and exposed to UV light. This exposure provided an area of pure silicon on each sample for THz transmission references. NiTi alloy layers were then

69 55 deposited via Ar-plama assisted chemical vapor deposition (CVD) with an AJA International Orion 4 Magnetron Sputter system using parameters determined in previous experimental work done by the Koretsky group. For more details of the capabilities of this system, see reference [96]. The samples were not annealed, and hence the structure of the NiTi alloy was amorphous [66]. The Ti% composition was determined using energy-dispersive x-ray spectroscopy (EDX). When bombarded by an electron beam, an EDS detector measures the energy and quantity of subsequently emitted x-rays. The x-ray measurements allow the atomicresolution mapping of the composition of large areas of materials [97]. Figure 5.1 below is an example of one of the spectra used to determine the composition. Fig. 5.1 Sample EDX spectrum used to determine Ni and Ti concentration.

70 56 The thicknesses, d, of the deposited films were determined by atomic force microscopy (AFM). Figure 5. shows several measurement data sets. Besides being able to determine the thickness of the film, these measurements show that beyond ~ 1 μm from the edge, the films are uniform. Fig. 5. AFM measurements of the thickness of several NiTi alloy thin films. The model developed later in this chapter to analyze the experimental results is quite sensitive to the thickness of the films, so, as we will see, accurate measurement of the thickness of the NiTi alloy films is crucial to an accurate calculation of the resistivity. 5. Experimental Setup The experimental setup is depicted below in figure 5.3.

71 57 Fig. 5.3 Depiction of NiTi sample and detection schemes. (i) Power transmission with a Si bolometer and (ii) Field amplitude transmission with EO sampling. The source of the THz pulse is the same as outlined in sections 3.1 and 3. for broadband, single-cycle THz pulses. Each THz pulse is focused onto either the alloy or the substrate. The beam diameter at the focal plane is ~.5 mm. The consequent transmission is measured either with the bolometer or the EO sampling setup depending on the type of measurement being done. As mentioned above each sample was deposited on a Si substrate (3 μm thick) such that a portion of the sample remained bare Si. The Si area ensured that the reference for each film was from the same portion of wafer for consistency. As mentioned in section 3.4.3, this reference is important for extracting the properties of the thin film without having to worry about possible EO sampling pulse distortions from the substrate or detection crystal.

72 Transmission Measurements The first measurements done were THz transmission images generated by raster scans of several samples. These earliest samples did not have reference Si available on the same portion of wafer. Instead, a separate piece of the wafer these samples were grown on was used a reference. Although this is slightly less rigorous than the method adopted later, some important preliminary results were gained. All samples, including the reference, were attached over 5 mm wide circular holes in a thin aluminum plate. An example of these Raster scans is shown below in figure 5.4. d = 3 nm d = 6 nm T.8.6 mm mm.4. Fig. 5.4 Raster-scan of two NiTi alloy thin films of different thickness on Si substrate. The pixel size in these images is 4 μm with a pixel integration time of 1 ms. These images demonstrate the expected increased absorption by the thicker film. More

73 Rel. Trans. ( E NiTi+Si / E Si ) 59 importantly, within each sample, the transmitted power is uniform which reveals that the samples are spatially homogeneous on the scale of the THz wavelength. Similar power transmission measurements were taken on all subsequent samples and references. Although spatial homogeneity was present, we averaged over several measurements at various locations on the sample to be thorough. With these measurements, we can plot the relative power transmission ( R Pow E NiTi Si / ESi ). An example of some data is shown below. Ti % T Ni-Ti / T Si Titanium % Fig. 5.5 NiTi alloy Relative power transmission measurement using a Si bolometer. Figure 5.5 shows a relatively linear increase in transmission up to around 5% titanium. We can qualitatively justify this by the following hand-waving argument. Pure metals are likely to have a large number of free carriers, and hence absorb radiation efficiently for frequencies below the plasma frequency. As the alloy percentage is increased, often

74 6 the number of carriers decreases, and the number of scattering sites increases. Lowered free carrier concentration and increased scattering leads to less efficient absorption and a decrease in resistivity. In fact, many pure metals often have smaller resistivity than alloys of those metals [98]. The next data taken was THz field amplitude measurements using THz-TDS. Figure 5.6 shows a representative set of transmission waveforms for various Ti concentrations along with their corresponding relative transmission (R TDS = E NiTi+Si (ν)/ E Si (ν)) as a function of frequency. Despite the fact that these samples are of finite thickness, and are not uniform, there is essentially zero difference in phase of each of the transmitted waveforms (i.e. peaks and valleys occur at the same times). Fig. 5.6 NiTi alloy time-domain data. (a) Transmitted waveforms and (b) Relative transmission for various Ti concentrations.

75 61 As in the power transmission data, the general trend of figure 5.6(a) is for the transmitted amplitude to monotonically increase as the percentage of Ti increases. However, further investigation revealed a far more complex dependence on Ti concentration which will be shown in the next section. The curves for t rel in Fig. 5.6(b) were generated by performing a Fast-Fourier-Transform (FFT) on the time domain data. This figure displays a characteristic flat spectral response for all the samples. Some of the implications of the flat response will be discussed in the next section. 5.4 Analysis and Results Justification for the Theoretical Model In order to evaluate the experimental data, we developed a surprisingly simple theoretical model based upon treating the alloy film as an optically thin conductive film, and the Si substrate as an optically thick dielectric. (By optically thick/thin, we mean that the thicknesses of the film and substrate are respectively small and large when compared with the wavelength of the THz radiation). We applied these assumptions using a combination of the Drude Model DC conductivity and the Thin-Film Fresnel coefficients. There were several justifications for using this simplified model that were borne out by our experimental results. First of all, from a conceptual standpoint, pure metals in general have a high density of free carriers. Under many circumstances, these carriers act in a manner consistent with a kinetic theory such as the Drude model. In fact, for many materials, the Drude model can still provide useful first order information [99,1]. Despite the fact

76 that alloys will in general have a lower density of free carriers than their constituents, they still have large number, and we can expect to see Drude-like behavior. 6 Another justification for this theoretical framework comes from a consideration of the scattering times in metals. A major strength of the Drude Model in predicting qualitative and quantitative results is that you can derive results related to scattering without the need of specifying the scattering mechanisms themselves [11]. All that is necessary is to assume there exists some scattering mechanism, and specify the scattering time, τ s, which is the average time elapsed between scattering events. In the case of metals, this is usually on the order of 1-14 seconds at room temperature [11]. On the other hand, from figure 5.6(a), we see that the THz pulse duration is much longer than this and is given by τ pulse ~ 1-1 seconds. In other words, the frequency components that make up the THz pulse do not extend high enough such that the oscillations of the THz field are fast enough to act on the carriers on a time scale that is relevant compared to the scattering time. The carriers are scattered many times before the THz field changes appreciably and therefore have no memory of the THz field. The relative smallness of the scattering time with respect to the THz frequency is one indirect reason that the transmission has a flat spectral response. The carriers in the NiTi essentially see a time varying DC electric field, and hence we can calculate the DC resistivity. How the DC resistivity comes about in our calculations will be developed in the next section. As stated above, one final justification for this formulation lies in the relative thicknesses of the alloy film and the substrate. In general, the reflectance and transmittance of an electromagnetic wave at a conducting surface is a function of

77 frequency. However, in the case of the NiTi, we can neglect this by virtue of the fact that the thickness of these films is many times smaller than the skin depth of this material. To check this, it is simple to calculate the skin depth from the Four-Point-Probe values for the conductivity of the thin films. After the initial Raster scans, the thickest films were on the order of 8 nm. (By comparison, the peak amplitude THz frequency generated in our single-cycle arrangement is about 1 THz. 1 THz translates to a wavelength of min THz 3 m ). A simple formula to calculate the classical skin depth, δ, is given by [1] 63 (5.4.1) f Applying values for the approximate smallest value of resistivity, a frequency of 1 THz, and magnetic permeability we find that 3e 7 m 76 nm (5.4.) 1 (4 ) (1e 1 s ) (1e 7 s) giving a ratio of d / 3. 45, where d is the film thickness. The ratio shows that the films are indeed much less than the skin depth of the metal. This relationship is also responsible for the flatness of the Relative Transmission spectrum which was observed. The Si substrate had a thickness of 3 μm. Combined with an index of refraction that is essentially constant at n = 3.418, the optical path length in the substrate for the THz is OPL nl ( 3.418) (3 m) 1. 5mm (5.4.3)

78 64 showing that the width of the substrate is much larger than the THz wavelength. Additionally, there is virtually no absorption of THz in Si at these wavelengths, so treating it as a dielectric is justified [13,14] The Drude Model The Drude Model is a kinetic theory applied to the electrons in a metal to describe carrier dynamics. This model still yields valuable qualitative and quantitative information about many metal systems despite its seeming naivety. The reason for the model s continuing viability is due to the fact that metals often have a high density of conduction electrons that essentially act as free carriers. Alloys usually have a lower carrier density than pure metals, but often still have a high enough density to be treated with the Drude model. Using the Drude model, we can directly relate electrical response properties (conductivity/resistivity) to a few simple quantities such as carrier mass (m), charge (e for electrons), density (n), and scattering time (τ s ). In the Drude model, this is expressed through Ohm s law as [11]: J E with ne m s (5.4.4) In order to connect this model to the Fresnel Thin-Film coefficients, we need to relate the conductivity to the index of refraction of the metal film. The connection is forged by starting with Maxwell s equations (.1.1-4). As usual, we take the curl of equation.1.1 and insert equation.1.. Applying a more generalized Ohm s law for now, J( ) ( ) E( ), yields

79 65 t E E t E ) ( (5.4.5) If we consider the incoming field as t i e E E this expression becomes E c i E E i E 1 ) ( ) ( (5.4.6) A direct result of the Drude model is that the frequency dependent conductivity can be written as [11] s i 1 ) ( (5.4.7) The product ωτ s is small enough in our thin films that we can neglect this portion and approximate σ(ω) as the DC conductivity, σ. Armed with this approximation, and some knowledge of the wave equation, we can rewrite equation as 1 ~ ~ E i E E c n E E k E (5.4.8) From equation 5.4.8, we can write the complex index of refraction as a function of the DC resistivity as follows 1 1 ~ ~ i i n r (5.4.9)

80 Equation will be used in the next section to apply the Drude conditions to the Fresnel thin film equations Thin Film Fresnel Coefficients Since we are treating the NiTi alloy as a thin conducting film, we are primarily concerned with what happens at the surface that defines the interface of the alloy and the substrate. As we will see, this means that the alloy film acts as an infinitesimally thin absorbing interface that has very little effect on the phase of the propagating THz pulse. To calculate how the sample effects the transmitted THz, we first consider what happens at the Si/NiTi interface using the figure below. r 3 t() t1 r 1 i t 3 e t 3 e i3 n 1 n n 3 n 4 r 3 r 1 t 3 r 1 t r() 3 3 r 3 e i i4 n 1 n n 3 n 4 t e (a) (b) Fig. 5.7 Illustration of the interface of NiTi alloy and silicon. The darker section on the left (n ) is NiTi alloy, while the lighter right-hand section (n 3 ) is silicon. (a) The total transmission, t(ρ), through the interface, and (b) the total reflection, r(ρ), from the interface.

81 67 Because of the thinness of the films, we must sum up all transmissions and reflections at the interface. Figures 5.7 (a) and (b) show the first few in each sum respectively. Note that the experiment is performed at normal incidence, and an angle has been added to the incidence and internal transmissions/reflections for illustrative purposes. Each time an internal transmission or reflection traverses the film, a phase is added to the propagating wave defined by nd, ( d film thickness) (5.4.1) c This phase is not negligible, but still very small. In essence, the absorptive effects of the carriers in the film are far larger. The coefficients, t ij and r ij, are simply the Fresnel transmission and reflection coefficients [15] at normal incidence given by t ij n i, n n i j r ij n n i i n n j j (5.4.11) Using figure 5.7, we sum up the total transmission and reflection across the interface. The two sums are given by: i i n t( ) t1t3e [ r3r1e ] n i r( ) r3 t3r1t3e [ r3r1e n n, 1, (# of reflections ) i n ] (5.4.1)

82 68 Since the ratio λ THz /d ~ 35 ~ n, the sums above can be simplified by the identity n 1 x. Applying this identity, we arrive at the following exact expressions for 1 x n the transmission and reflection t t t e r r e (5.4.13) i i ( ), r( ) i 1 r1r3e 1 r i 1r3e By applying some approximations, we can further simplify these equations. The first is that because of the large absorption demonstrated in the transmission measurements, we assume that n n1, n3. Secondly, although n is large, ( nd) is small because the c thickness, d, of the NiTi alloy is so small. We cannot drop the phase altogether, but we can apply the small angle approximation through Euler s formula. Lastly, we assume that the absorption is by far the dominant effect of the NiTi alloy film. This assumption i manifests itself in the relationship 1 and thus, ~ i n from equation By inserting equations , applying these approximations, and performing a bit of algebra, we arrive at the simplified expressions t( 1 n 3 n3 n1 dz n3 n1 dz 1 ), r( ) (5.4.14) n n dz where Z is the impedance of free space. Notably, as expected from our justifications for this model as well as the experimental data, these functions are independent of the THz

83 69 frequency. Thus, we have expressions in our model that completely describe the interface between the NiTi alloy and Si and are only functions of the DC resistivity. Equipped with these formulas, we can explore the functional dependence of the THz field transmitted out of the Si substrate which is what we are actually measuring. We do this in a similar fashion using figure 5.8 below as a guide. Fig. 5.8 Illustration of the THz transmitted out of NiTi sample. (a) NiTi on Si, and (b) the Si substrate. Again, we account for the initial transmission followed by a series of internal reflections. By comparing the transmission through the NiTi+Si to the transmission through the bare

84 Si substrate, we can derive an expression for the relative transmission, R = t NiTi+Si / t Si, which is what was measured in section As the measurements performed by the bolometer are a time-integrated measurement, we need to account for all the subsequent internal reflections depicted in figure 5.8. Summing all transmissions leads to the following expressions: T T NiTi Si Si t Si t NiTi t Si t NiTi t Si t NiTi t Si t NiTi t t 1 t t r31r34 34 r r34 (5.4.15) Dividing these two expressions by each other yields the relative transmission R Pow T T NiTi Si Si t t 13 1 r 1 r r r34 (5.4.16) As t and r are a function exclusively of ρ, we can use this equation to solve for the resistivity. The resistivity is calculated by using a least-squares approach to fit the formula to the measured values of R pow. For the THz-TDS data, a similar approach is used. However, in this case, we are mapping out the field in time, and can see the individual pulses as they arrive at the EO sampling detector. Figure 5.9 demonstrates a few longer THz-TDS measurements which show the initial pulse and subsequent decaying transmissions.

85 Amplitude (V) 71 (a) (b) tniti t NiTi t NiTi t Si t Si t Si Time (ps) Time (ps) Fig. 5.9 Long TDS measurements taken on NiTi alloy. These long scans show the initial transmitted pulse and two subsequent internal reflections of (a) the NiTi alloy on Si, and (b) the Si reference. As is shown in figure 5.9, each pulse is distinct in time. Also, we notice a few effects that demonstrate the strong optical response of the NiTi alloy film. First, each successive pulse is heavily attenuated when compared with its constituent reference pulse. Secondly in figure 5.9(a), we notice a π phase shift in the polarization of t and t not present in figure 5.9(b). This phase shift is due to the large index of refraction of the alloy film. Since each pulse is distinct in time we can examine each one individually and extract the resistivity from wherever we choose. Again, using figure 5.8, we can write an expression for the n th pulse (n =, 1,, with n = corresponding to t, etc.): t t ( n) NiTi Si ( n) Si t tt t ( rr ( r r 34 ) ) n n e e i(n1) i(n1) (5.4.17)

86 Since the first transmitted pulse has the highest signal to noise ratio, we choose this set which gives the following simple expression for the relative transmission: 7 t t () R TDS (5.4.18) 13 Again, after measuring R ρ. () TDS, we can use a least-squares best fit to determine the value of Results With the analysis above, we calculated the DC resistivity of several samples ranging in Ti concentration from -1% using both the power transmission measurements and the TDS transmission values. Additionally, as mentioned above, we also performed Four-Point probe measurements. The results are below in figure 5.1.

87 73 Fig. 5.1 Summary of resistivity measurements made on NiTi alloy thin-film samples. There are several peaks located near the titanium concentrations of %, 44%, and 6%. The dashed line is a fit based on an effective medium treatment. Figure 5.1 show the dependence of the NiTi alloy thin film resistivity on Ti concentration. As can be seen, there is good agreement between all methods of measuring ρ, indicating that THz wave transmission is a reliable method for extracting the carrier properties of these thin-films. The dashed line on the graph represents a first approximation of the resistivity based upon a very basic effective medium treatment. eff This first approximation assumes that the effective resistivity, NiTi, is purely a linear

88 74 combination of its constituent parts ( Ni and Ti ), weighted by concentration (x). In the event that the structure of the constituents is such that the pure domains of Ni and Ti are larger than the mean-free-path of the carriers, the plot should reduce to this form. That is clearly not the case in figure 5.1, which implies a structure much more complicated than this. However, the measurements at either end (pure nickel, and pure titanium) agree with this by definition. The most striking features of this plot are the peaks at, 44, and 6% respectively. These peaks indicate a drastic increase in the resistivity of the films. Although we cannot be certain, we speculate that these peaks correspond to an increase in the disorder of the structure at these concentrations. This speculation is supported by figure 5.11 showing phase transitions which occur at these concentrations [16]. Fig Phase diagram for NiTi alloys. This figure illustrates that the material undergoes phase transitions at Ti concentrations corresponding to increases in resistivity in our measurements.

89 75 In section 5.1, it was mentioned that the samples were not annealed, and hence are amorphous. However, the samples seem to still maintain some vestige of these low temperature phase transitions as they near specific concentrations. If this is indeed the case, THz spectroscopy might be used as not only a tool for extracting electrical properties, but also as a rudimentary tool for mapping of phase transitions. Although the wavelength of the THz pulse is too large to be sensitive to the small domains in the alloy, it may still be sensitive to them through their effects on the carrier dynamics. In conclusion, we have systematically studied the carrier dynamics of NiTi alloy thin-films by examining the DC resistivity using THz imaging and THz-TDS. These studies allowed us to map the resistivity as a function of Ti concentration. These calculations yielded a surprisingly complex structure that indicates several concentrations at which the resistivity increases. We speculate that at these concentrations, the disorder of the film is greatly increased due to being near to a phase change, causing increased scattering and hence the spike in resistivity. This work was accepted for publication in Applied Physics Letters in May, 11 [17].

90 6. Transient Optical Response of Quantum Well Excitons to Intense Narrowband THz Pulses 76 As most semiconductors have band gaps on the order of E gap 1 ev, THz photons do not excite inter-band transitions. However, they are often ideal for exciting intra-band transitions as these transitions are often of the order of a few mev (1 THz ~ 4.14 mev). Because of this, the interaction of condensed matter with THz radiation is of fundamental interest for a variety of reasons. These include carrier transport properties such as Coulomb effects and many-body interactions. These carrier dynamics can manifest themselves in a variety of nonlinear effects such as the Dynamical Franz-Keldish effect [18-11], the Quantum Confined Stark Effect [ ], Carrier Wave Rabi Flopping [ ], and ponderomotive effects [118,119]. Additionally, there are important practical research applications such as the development of ultrafast communications beyond the GHz [1-1], and quantum information processing [13,14]. The excitation of intra-band dynamics with THz photons is particularly interesting because several effects which normally are not of the same order now have similar energy. These include the ponderomotive energy of the carriers, the Rabi energy of the carriers, and the modulating photon energy. These experiments are largely accessible because of the prevalence of high-power, ultra-short, pulsed laser systems. The THz outputs accessible with these systems allow us to enter the regime of extreme nonlinear optics, which bears some differences from ordinary nonlinear optics. Some important differences include the breakdown of a perturbative treatment, the failure of the rotating

91 77 wave approximation (RWA), and the importance of non-resonant effects which can lead to Rabi oscillation of the Bloch vector [15]. In addition to these differences, there are a host of extreme nonlinear transients that can be observed. These nonlinear transients have been demonstrated with a single-cycle, broadband THz pulse [118]. In this study, we resonantly excite excitonic polarization and strongly couple the intra-band states with a narrowband THz pulse. What exactly this entails will become clearer after some background information is provided. 6.1 Quantum Wells It is with this spirit in mind that we explored the interaction of THz photons with excitonic polarization in an AlGaAs quantum well (QW). A quantum well is essentially a material with a given potential energy surrounded by materials with a higher potential energy thus creating an area of confinement in the lower potential material. The confinement properties of QW s make them ideal nano-scale laboratories in which to discover the properties of carriers in semiconductors. Figure 6.1 shows a generalized 1- Dimensional QW. As depicted in figure 6.1, the confinement of carriers leads to discrete energy levels. Additionally, as we will see, the confinement also leads to stronger coupling between carriers.

92 78 Fig. 6.1 Simple depiction of a 1-dimensional quantum well. Higher potential (U ) barriers surround a lower potential (U 1 ) material forming discrete bound states in the well. Neglecting the momentum in directions not along the Z-axis, and consideringu U1, we can approximate the energy levels from the infinite potential well solution: E n z n z (6.1.1) m l e where m e is the effective mass of the carrier. Considering the dimensions of the QW to be the order of l = 1 nm, the first energy level n z = 1, and the effective mass to be that of an electron, we see that this energy falls near the THz photon energies generated by our sources (E 1 ~ 3.7 mev). This simplistic approach is not the correct formulation for the carrier energy as we will see in the next section, but it does demonstrate that we can scale

93 the energy levels of the quantum well by changing its dimensions. Specifically, we will be more concerned with the energy spacing between adjacent energy levels. 79 Our specific sample used in this study was a stack of 1 high quality, 1 nm thick GaAs QW s separated by 16 nm thick Al.3 Ga.7 As layers. The band structure of this QW is depicted below in figure 6.. Fig. 6. Bulk and QW GaAs band structure. Left hand side: bulk GaAs band structure. Right hand side: GaAs QW band structure when surrounded by Al.3 Ga.7 As. Splitting of heavy hole (HH) and light hole (LH) is caused by the strain introduced at the interfaces of the layers. When Ga and As are combined, the 4s and 4p shells are occupied with the outer electrons from the elements. The p-level is three-fold degenerate, with each level containing two electrons. These three p-levels comprise the highest levels of the valence band when the

94 8 GaAs is in a crystalline structure. The angular momentum configuration of the valence bands is important to the selection rules for optically excited transitions. Spin-orbit coupling breaks the degeneracy of these levels. The bulk GaAs on the left hand side of the figure 6. shows the top two J = 3/, heavy hole (HH) and light hole (LH), valence bands which are degenerate at the Γ-point. However the strain introduced by surrounding the GaAs QW with the Al.3 Ga.7 As barriers again breaks this degeneracy and shifts the levels of the valence and conduction bands. The broken degeneracy yields two clear transition possibilities which are evident in the absorption spectrum measured for this sample in the range relevant to our laser system. Fig. 6.3 Absorption spectrum of GaAs/Al.3 Ga.7 As QW used in our study.

95 81 Figure 6.3 demonstrates the quality of the sample in that it shows clear, sharp absorption resonances at wavelengths of approximately 796 nm and 85 nm. We will discuss in more detail the carrier properties that are involved in these absorption resonances. In particular, we will see that these resonances correspond to a different effective mass for the LH and HH. 6. Excitons In the previous section, I have been careful to restrict my description of the particles under investigation in the QW s to the vague term of carriers, or to electrons and holes. Now I will be more specific with a description of excitons, the quasi-particles involved in these experiments. When an electron is excited from the valence band of a semiconductor, it leaves behind an empty space. The empty space can be treated as a positively charged quasi-particle called a hole. The oppositely charged quasi-particles attract each other and, in short, an exciton is a bound state of the electron and hole held together by their Coulomb attraction. Because of this negative Coulomb energy, the excitation energy required to create an exciton is less than the band gap energy of the semiconductor. In optical spectroscopy, this can lead to strong absorption below the band edge [16,17]. To better understand how this quasi-particle will behave, we explore some of its basic properties as they occur in the GaAs/Al.3 Ga.7 As QW s described in the previous section. We realize right away that this system bears a similarity to the hydrogen atom

96 8 with a hole instead of a proton. In fact, the treatment for the free exciton is mathematically identical, and yields the following binding energy: E 4 mre 1 1 E B 3 n n Free Exciton (6..1) This energy level structure is the same as the hydrogen atom with a few exceptions. One noticeable exception is the inclusion of the background dielectric constant, ε, which is important for carriers in semiconductors, but is excluded for atomic hydrogen as ε = 1 for a free atom. The importance of the dielectric constant will be expanded once we have determined some exciton properties. Additionally, we use the reduced mass of the electron-hole combination, m r, expressed as 1 m 1 1 r me m (6..) h As we saw in section 6.1, the band structure depends heavily on whether or not one is working with bulk GaAs or a QW structure, hence the effective mass of the electrons and holes, m e and m h, will vary accordingly. The effective masses are especially important at k =, where the degeneracy of the LH and HH is broken by the QW barrier materials. For GaAs, the dispersion, and hence LH and HH masses are given in terms of the Luttinger parameters [18], γ 1 and γ, such that the LH and HH effective masses near k = are given by m m, m m LH HH (6..3) 1 1

97 83 We will use these to determine properties of the GaAs/Al.3 Ga.7 As QW s later, but first we need to determine the form of the bound state energy levels in our case. Following the procedure of Peyghambarian, Koch, and Mysyrowicz [18], we begin with the single exciton Hamiltonian: Coulomb confine h h e e V V m m H (6..3) Equation 6..3 consists, from left to right, of the kinetic energy of the electron and hole, a term for the potential due to the confinement barrier of AlGaAs, and a term for the Coulomb interaction of the electron and hole. As in figures 6.1 and 6., we assume the confinement direction is along the z-axis. We then separate the x-y plane coordinates from the z-axis, and make the following coordinate transformations:, y x Y X r R (6..4) In which X and Y are the center of mass coordinates, and x and y are the relative coordinates defined by h e h e e h e e h h e h e e h h y y y x x x m m y m y m Y m m x m x m X,, (6..5) Additionally, we use the x-y plane effective mass, m xy and the x-y plane total mass, M xy defined as

98 84 xy h e xy xy h e xy m m M m m m, (6..6) Making these changes we arrive at the Schrödinger equation ) ( ) ( r E r r e V m M z m z m confine r xy R xy h h e e (6..7) By defining h r e r r, the function ) (r can be separated into x-y plane components and z components. The solution for the motion in the x-y plane, which can be found in [19], yields the following expression for the bound energy levels for excitons 1 j B z r g D n n E L m j E E (6..8) From left to right, these terms are the band gap energy, the additional energy due to the confinement potential, and the exciton binding energy where E B is the excitonic Rydberg from equation Keep in mind that this solution is for an infinite potential barrier, as indicated by the confinement energy in the second term, which is not the same as our QW, but will serve to demonstrate some important properties of the excitons. Comparing the last term in equation 6..8 to equation 6..1 we see that there is a factor of four difference in the ground state energy ( D D E E 3 4, for n = 1) caused by the reduced dimensionality of the QW system. Using this, and rewriting the two dimensional exciton binding energy as

99 85 E D (6..9) m r D a we can easily show that for the ground state, the Bohr radius of the -D exciton is half that of the 3-D case: a D m e r a 3D (6..1) With equation 6..1, we have quantified the intuitive property that the confinement due to the QW introduces a tighter binding of the electron and hole and a corresponding increase in energy. With the above treatment we can now calculate the LH and HH binding energies, as well as the two dimensional Bohr radius. We start by using equations 6..3 to calculate the LH and HH effective masses with respect to a free electron mass. Using the Luttinger parameters γ 1 = 6.9 and γ =.4 found in [18], we find the following values m LH.855 m, m.476m (6..11) 6.9 m HH m Using these values, the electron effective mass, m e =.67m, and equation 6.., we calculate the LH and HH exciton reduced masses: m LH r HH.376 m, mr. 587 m (6..1)

100 86 Finally, using a background dielectric constant of ε = 13.1ε [13], and putting these values into the expressions for the ground state binding energy and exciton Bohr radius, we obtain the following values: E LH Binding 11.9 mev (.9THz ) E HH Binding a a LH HH 18.5 mev 9.3nm 6nm ( 4.5THz ) (6..13) From equation 6..13, we see the binding energy falls into the THz range, and that the distance between the electron and hole is significantly larger than the GaAs lattice spacing of ~.5 nm. As mentioned before, we are mainly concerned with the spacing between the ground state and the first excited state. Using the HH, we find that the excitonic HH Rydberg, HH EBinding EB. Using this, we find the energy spacing between the ground and 4 first excited state to be E 1 4E B.44E B ev 4THz (6..14) This frequency is higher than the DFG system described earlier can produce; however, the actual energy spacing is smaller for a few reasons. First, although this treatment illustrates the properties of excitons fairly well, we must keep in mind that this is not a strictly two dimensional system. Because there is a finite, but significantly smaller,

101 87 dimension along the growth direction, the system is known as quasi-two dimensional. The small, but finite, size has the effect of lowering the binding energies and thus increasing the Bohr radius. Additionally, the potential barriers of AlGaAs are not infinite, but about.5 ev higher than the bandgap of GaAs. The finite barriers will result in closer spacing of the ground state and first excited state as well. An argument for this can be made by comparing the energy spacing of the infinite potential well in one dimension to the spacing of the finite one dimensional potential well. In this problem, the finite well has a potential barrier of height V. Both wells have a carrier mass, m, and well width, L z. I will not solve this problem here, but will simply state the solutions for the figure 6.3 below. The solutions for the energy levels are as follows [131]: Infinite Potential Well : E ( j) z j (6..15) j 1,,3 ml z Odd Finite Potential Well : Even : :- E z E tan z cot me me L z z L z z V V E z E z (6..16) The solutions to the above equations are represented graphically below in figure 6.3.

102 88 Fig. 6.4 Graphical depiction of the first two energy levels for both the infinite and finite barrier quantum well systems. Black curves represent the even solutions, while white curves represent the odd solutions. The dotted line represents the term V Ez common to both solutions. The asymptotes show the infinite well energy levels. As the solutions for the finite well are in the form of a transcendental equation, they cannot be solved for analytically. However, by plotting them in figure 6.3, we can see a distinct difference in the energy spacing of the first two levels for the different QW s. It is obvious that for similar materials and dimensions, the spacing between the first two energy levels in a finite QW will be substantially less than those for an infinite well. 6.3 Experimental Arrangement The experimental arrangement used is depicted below. With this, some knowledge of the QW in question, and the carriers we are interested in, we can discuss the specifics of the dynamics we probed.

103 89 Fig. 6.5 Experimental Arrangement for the DFG QW experiment. The THz wave generation for this experiment is described in detail in Chapters 3 and 4. Using the DFG source, we had fine and tunable control over the THz center frequency. Generating THz radiation with the DFG consumes most of the optical pump power (~%95), with the remaining 5% used for the optical probe. However, this setup requires us to stretch the optical pump in order to introduce the chirp necessary for this THz generation method. As such, there is an additional compressor on the optical probe line. The compressor consists of a lens, mirror, and high quality diffraction grating. Similar setups, sometimes using two gratings instead of a mirror, are commonplace for compressing chirped pulses [13]. The simple schematic below shows the compression process.

104 9 Fig. 6.6 Schematic of pulse compressor. High frequency components are depicted as gray. Low frequency components are depicted in black. The figure above demonstrates graphically how the leading high frequencies undergo a slightly increased path length due to the diffraction grating. The larger path length allows the lower frequencies to catch up to the higher frequencies, and thus recompresses the pulse. By tuning the distance, Δl, while observing the pulse duration with an autocorrelator, we can optimize the compression of a range of stretched pulses. After the probe is recompressed, the THz pulse and optical probe co-propagate and are focused onto the QW sample. The transmitted optical probe is then focused into a spectrometer. As the delay between the optical probe and THz pulse is changed, we observe time-resolved modulation of the transmitted optical spectrum, T(ω), due to the strong THz pulse. Specifically, we observe the THz-induced modulation of the LH and HH exciton resonances. The optical probe alone excites a resonant excitonic polarization at the first

105 91 excited state of the LH and HH excitons. Since the exciton energy levels are hydrogenic, they display similar angular momentum characteristics and are referred to by the same spectroscopic notation (s, p, d, f ). Hence, this first excited state is named the 1s state. As the symmetry of the LH and HH states are p-type, and photons only deliver one unit of angular momentum, transitions from these states to the first excited p state are not allowed. As a result, the p state is known as an optically dark state. However, a two photon process is capable of reaching this state. As described in the previous section, the energy separation between the 1s and p state falls near our THz photon energy. Coupling an appropriate narrowband THz pulse to the 1s excitonic polarizations, we can excite a resonance between the 1s and p states. The resonant transition between the 1s and p states is depicted below. Fig. 6.7 Internal structure of confined excitons. This diagram illustrates the excitation of the 1s excitonic polarization and the subsequent strong coupling of the 1s and p states by an intense THz pulse.

106 9 The spacing of the 1s and p levels are exaggerated for clarity. We know from the previous section that the levels are within a few mev of the conduction band. Therefore, it is also easily possible to ionize the electron into the conduction band if the THz frequency is large enough. It is important to note that this is not a weak, linear process. There is a strong coupling between the light (THz photon) and matter (exciton) that cannot be treated perturbatively. However, analogous with the perturbative treatment of a harmonically driven two level system, the driving field causes a cyclic rotation between the two energy levels known as Rabi flopping. Rabi flopping is characterized by the Rabi frequency, Ω R, defined by: E E ) (6.3.1) R p 1s ( p 1s Additionally, a general result of classical mechanics indicates that strongly coupling two oscillators will result in a splitting of the Eigen-frequencies of the system [133]. The splitting that occurs is also observable in quantum mechanics, where strongly coupling two well-defined quantum states will result in a splitting of the energy Eigen-states in the following manner: 1 1 i, c 1, c e,1 (6.3.) Where 1 and are the original states, 1, and, 1 are the strongly coupled states, c 1 and c are the probability amplitudes, and e iφ is the definite phase between the two states. Including the coupling-induced splitting, the following figure details the dynamics of this experiment.

107 93 Fig. 6.8 Depiction of exciton polarization dynamics in our QW system. Figure 6.8 illustrates that the weak optical probe excites an excitonic polarization at the 1s frequency. The subsequent intense THz pulse strongly couples the 1s and p states which results in Rabi oscillations, and a splitting of the 1s and p level. The symbol g represents a weighting factor that is proportional to the detuning of the THz energy from the Rabi energy, E p-1s. (For example, if Ω R = ω THz, g =1). Figure 6.8 suggests a simple way to find if this strong coupling occurs. In the absorption spectrum of our QW (figure 6.3), we see two distinct absorptions from the LH and HH. If this strong coupling regime is present, and Rabi Oscillations occur, we should see evidence of the absorption spectrum splitting into two distinct sidebands.

108 Results Experimental Results The results of this experiment compare experimental observations and theoretical calculations of strong interactions between intense THz radiation and the 1s-to-p transition of the excitonic polarization in resonantly driven GaAs/AlGaAs QWs. The theoretical calculations were performed by our collaborators in the Koch group in Marburg, Germany. These calculations, which will be briefly discussed later, are extremely complicated many-body calculations which include microscopic Coulomb effects, and THz effects. As we will see, there is an excellent agreement between the observations and theoretical results. The experiment was a time-resolved, THz-pump/optical probe experiment. The light source was 8 nm, 9 fs pulses from a 1 khz Ti:Sapphire regenerative amplifier (Coherent, Inc, Legend). The THz pulse energy was in the range of 1 nj, and the pulse duration was ~3 ps. The maximum electric field amplitude reached approximately 5 kv/cm. Figure 6.9 (a) and (b) shows a TDS measurement of a typical multi-cycle pulse generated by the DFG setup for this experiment and its FFT spectrum. Figure 6.9 (c) shows the 1-T(ω) optical spectrum at a single time delay out of the entire time resolved scan.

109 95 Fig. 6.9 Modulated 1-T(ω) optical transmission spectrum. (a) TDS measurement of DFG pulse with (b) its FFT spectrum centered at ~1.86 THz. (c) 1-T(ω) optical transmission spectra at t d =.6 ps. The light line is the un-modulated spectrum. The dark line represents the THz modulated spectrum. The light line in figure 6.9(c) shows the un-modulated optical 1-T(ω) spectrum. This spectrum represents the linear dipole transition strength from the LH and HH valence bands to the 1s exciton state. The dark line shows the 1-T(ω) spectrum modulated by an intense multi-cycle THz pulse. Immediately several effects are noticeable. The peak amplitude of the spectrum is heavily attenuated, and both peaks are broadened and redshifted. In addition, as the arrows indicate, we see the onset of an additional peak on the high-energy side of the HH peak. These effects already suggest very strong non-linear

110 96 effects. To further characterize these effects, we measured the transmission at several frequencies ranging from THz. Figure 6.1 shows a representative sample of these modulated transmissions at various values of Δ where THz 1s p HH (6.4.1) In other words, Δ represents the difference between the driving THz photon center frequency, and frequency corresponding to the HH 1s-p transition energy of 1.75 THz).

111 97 Fig. 6.1 Several 1-T(ω) spectra. (a) THz excitation below the 1s-p HH resonance. (b) THz excitation near the 1s-p HH resonance. (c) THz excitation above the 1s-p HH resonance. The shaded areas are the un-modulated transmission spectra, and the dark lines are the modulated spectra. Insets show the THz excitation spectrum. Δ = ν THz ν 1s- p. These three THz excitation frequencies are representative because they correspond to excitation below, at, or above the 1s-p HH resonance. In all three cases, large nonlinear modulations of the spectrum occur. In the case of Δ = -.3, we already see a strong

112 98 broadening, along with a slight red-shift, and a lowering of the peak transmission. As Δ approaches zero, these effects become even more pronounced, and we again see the subtle arrival of a second high-energy shoulder that appears to be the onset of Rabi splitting. As the resonance moves to Δ =.4, nonlinear modulations are still very strong despite being off resonance. These strong modulations are present because the THz spectrum can now reach higher exciton states and into the conduction band continuum. The disassociation of excitons into a correlated electron-hole plasma leads to a rapid decoherence of that portion of the exciton population which washes out the coherent Rabi oscillations in the spectrum while still demonstrating a strong response. Below we see the full time evolution of the QW experiment of which figures 6.9 and 6.1 represent snap-shots in time. Fig Full time evolution of differential transmission. This differential transmission (ΔT = (T Opt+THz T Opt ) / T opt ) shows the THz effects on QW system with corresponding THz pulse for reference.

113 99 Figure 6.11 depicts the differential transmission of the system which is another very useful way to view the data. The differential transmission, T, is the THz-pulse modulated signal minus the optical signal alone divided by the optical signal again for normalization (See caption in figure 6.11). The plot in figure 6.11 reiterates a few important points. First, the dashed line falls across spectral oscillations occurring at negative time delays. These oscillations are a hallmark of coherent polarization because the pump and probe do not overlap in time at this point. The reason we see an effect is that the optical pulse excites a coherent interacting excitonic polarization as it arrives first. As non-interacting two-level systems, such as atoms, do not display this behavior [134], we know that the excitons have a definite phase relationship between their polarization oscillation frequencies (which is what is meant by coherent in this case). The coherence time of this system lasts until the THz pulse, with which it interacts, arrives later. Additionally, we can see the time development of the splitting of the HH into a red-shifted peak (left arrow fig. 6.11) and higher energy shoulder (right arrow fig. 6.11) that appears to be an effect of Rabi oscillations Theoretical Results and Comparison In order to explain these nonlinear effects, we compare the experimental observations with theoretical calculations. I will only outline a few preliminary steps and procedures in the theoretical treatment to illustrate its basis, but I will not go into the calculations as the complexity of the details [114,135] are outside the scope of my expertise. We begin with the system Hamiltonian:

114 1 THz PHON C SYS H H H H H (6.4.) H is a non-interacting portion, H C is the carrier-carrier interaction term, H PHON is the carrier-phonon interaction term, and H THz is the THz light-matter interaction term. As the experiment is performed at ~5K, we can neglect the phonon interaction contribution. Explicit forms for the other terms are given by: k k k k a a H,,, (6.4.3 a),,,,,,, 1 q k k q k q k k k q C a a a a V H (6.4.3 b) k k k THz k k k k THz THz k THz a a A D a a A m e A j H,,,,,,,, (6.4.3 c) In all of the above equations, a,k represents the creation/annihilation operator for an electron at band λ, and momentum k. Equation (a) is simply the kinetic energy of each carrier at momentum k and band λ. Equation (b) represents the carrier-carrier interaction with V q as the bare Coulomb potential of the system. Finally, (c) represents the THz interaction. The first line in represents intra-band transitions (λ i = λ f ) which contains linear ( t E A THz THz ) and nonlinear ( A THz ) THz effects. The last line represents THz inter-band effects. These effects are negligible compared to the intra-band effects as the THz photon energy is much smaller than the band gap [135].

115 11 Beginning with this Hamiltionian, one can derive the transmitted and reflected electric fields as a function of the THz current, J THz, from the wave equation. The transmitted and reflected electric fields can be used to write the Heisenberg equations of motion (EOM). The EOM s can then be solved in the Bloch basis which allows us to extract the exciton dynamics. Even from this rudimentary explanation, we can see that these EOM s involve an extremely complicated correlated, many-body calculation. However, an important detail that can be seen from even this cursory glance at the theory is that the dynamics are an explicit function of the THz electric field, E THz. In fact, the theoretical group uses the experimental form of the THz pulse in their calculations in order to represent the dynamics as accurately as possible. Below we see a sample of theoretical calculations that correspond to figure 6.1.

116 1 Fig. 6.1 Theoretical calculations of 1-T(ω). Shaded areas are the un-modulated spectrum. Dotted lines represent a calculation in which the p dephasing constant is artificially reduced. Solid lines represent the full computation. These calculations represent snapshots of the theoretical 1-T(ω) spectra at time delay t d =. The shaded areas are the un-modulated spectra. The solid line is the full calculation of the THz modulated spectrum which matches very well with observation, especially in that there is not a clear Rabi splitting. Instead we see the development of a shoulder as in the experimental data. The lack of clear splitting is due to the very rapid dephasing of the coherent polarizations. From this data, however, we were able to extract the polarization

117 13 dephasing constants of the excitonic 1s and p levels: γ 1s =.5 mev and γ p = 1.5 mev, which, to our knowledge, was the first time this had been done with this particular system. In fact, the dephasing constant for the p level is three times that of the 1s level. Returning to figure 6.1, the dotted line represents a calculation in which γ p is artificially reduced such that γ p = γ 1s. With this reduction, we see a much clearer Rabi splitting. This artificial splitting indicated that the onset of the shoulder is indeed due to Rabi oscillations, but that the very fast p dephasing constant prevents direct observation of the splitting. In fact, the calculations suggest that we could observe a splitting in the LH resonance as well since its 1s-p transition energy is also close to the THz photon energy. In conclusion, we have demonstrated that strong THz pulses can be applied to generate remarkably large changes to the optical response due to Rabi sidebands. Our analysis identifies the p-exciton dephasing to be three times larger than that of the 1sstate. Thus, the p dephasing is the limiting factor for the detection of pronounced sidebands. In order to increase resolution, it may be necessary to have samples with narrower exciton absorption lines to accomplish the reversible Rabi-flopping regime in GaAs-based QWs. Additionally; shorter probe pulses would be advantageous to increase resolution. Nonetheless, the THz pulses used produce pronounced nonlinearities even when full reversibility is not reached. This work was accepted for publication in November 9 by Applied Physics Letters [136].

118 7. Extreme Nonlinear THz Transients in Quantum Well Microcavities 14 Microcavities are excellent systems in which to study the interaction between light and matter extending from the weak-coupling regime to the strong-coupling regime. Because of this, much work has been done with atom optics in microcavity structures [ ], as well as reduced dimensional structures in microcavities [ ]. The non-perturbative exciton-photon coupling in a high-q microcavity leads to the formation of two new eigenstates called exciton-polaritons which will be discussed in more detail in the following sections. The splitting developed by the strong coupling of excitons to a microcavity is often considered a solid-state analog of vacuum-field Rabi splitting (~ΩR) of an atom-cavity system. Many fundamental questions concerning quantum optical phenomena in semiconductors have been explored by studying the optical properties of semiconductor microcavities. In the previous chapter, it was discussed that THz photons can excite states in coupled light-matter systems that are energetically inaccessible to optical spectroscopy. Indeed, the same is true for QW systems confined in microcavities. 7.1 Microcavity General Characteristics A microcavity is simply a resonant structure that can confine photons in a standing wave pattern. This structure is formed by having two highly reflective mirrors spaced by a distance, L. For any resonant cavity of length L, the number of resonant

119 wavelengths, or modes, is infinite, and corresponds to the number of half wavelengths of light that will fit between the two ends. A simple depiction of this is shown below. 15 Fig. 7.1 A generic resonant cavity. Available modes depend only on the optical path length (OPL) of the cavity, n L, where n is the index of refraction of the cavity. The field distribution of the standing wave inside the cavity above shows five antinodes (m = 5). The placement of the antinodes for any given mode is important since the field will be maximized at these points. Therefore, the antinodes are the optimal position at which to place a sample to observe strong coupling between the field and the material. Waves that do not fit into the cavity in half wavelength increments develop a destructive phase relationship, and are damped out very rapidly. As mentioned before, there are an infinite number of available modes. As the equations accompanying figure 7.1 indicate, these are all equally separated modes. The

120 mode spacing is sometimes referred to as the free spectral range (FSR) and is given by [149]: 16 c FSR (7.1.1) n L The actual modes supported are determined by the overlap of the available modes and the photons that are pumping the cavity. Since FSR is determined by the size of the cavity, how many modes overlap with the pump bandwidth can be controlled. For a Gaussian pump distribution, the overlap might look as follows: Fig. 7. Gaussian gain profile. (a) Overlap of Gaussian pump radiation profile with cavity modes. (b) Gain profile of cavity modes. Part (a) of figure 7. shows the overlap of a few modes with the Gaussian profile of the pump radiation. Part (b) demonstrates the actual gain profile of the modes. By controlling either the bandwidth of the pump radiation (FWHM), or the cavity size (L), or both, how many modes are amplified by the cavity can be controlled.

121 17 7. Polaritons When a material polarization is strongly coupled to a photon mode inside a cavity, a new quasi-particle is introduced called a polariton. In our case, this is an excitonpolariton. The optical photons simultaneously excite an excitonic polarization and excite cavity modes that couple to each other. The outcome of this coupling is similar to the situation described in chapter 6 in which the strong coupling of the 1s and p exciton states can induce a Rabi splitting of these energy levels. When the cavity resonance is near the excitonic binding energy, there is a strong coupling of the material state to the cavity state. The enhanced field produced by the cavity confinement induces a splitting of the original exciton state into two polariton states. An oversimplified, but illustrative, way to view this is to look at the dispersion relations of a single photon mode, and a single free exciton. Details of this method can be found in the dielectric treatment of polaritons found in chapter 11 of [19].

122 18 Polariton Dispersion ck n E(k) HEP k m ex const. LEP k Fig.7.3 Polariton dispersion relation. White lines show uncoupled photon mode k, and exciton mode k for small values of k. Black lines show the coupled modes giving rise to the lower Low Exciton Polariton (LEP) state, and High Exciton Polariton (HEP) state. The white lines in figure 7.3 show the uncoupled photon and exciton modes. For small values of k, the parabolic dispersion of the free exciton is approximately flat. When the modes couple to each other, the new quasi-particle forms two new states: the high and low exciton polariton states (HEP and LEP). For each value of k, there are two energetically distinct states possible. An interesting property of these new states is that they are neither photonic nor excitonic, but a combination of the two. For example, at the value of k where the dispersions cross, the LEP and HEP are an equal mixture of photonlike and exciton-like states. However, as you move to a higher (lower) value of k, the LEP state becomes more exciton-like (photon-like), and the HEP state becomes more photon-like (exciton-like). The balance of photon-like and exciton-like characteristics is

123 19 an important factor in considering the THz interaction with polaritons because THz photons should not interact with the cavity modes, but do interact with excitons. Hence, we expect the THz photons to have the largest effect on exciton-like polaritons and less effect on photon-like polaritons. 7.3 Our Sample The sample used in our experiment was a stack of 1 InGaAs QW s surrounded by an 11λ/ cavity consisting of Bragg reflectors designed to be %99.94 reflective. The figure below depicts the structure of the sample. Notice that the sample is wedge-shaped. (This is highly exaggerated in the figure). As was discussed in the previous section, the cavity resonance is only dependent on the OPL of the cavity. The wedge shape allows us to tune the fundamental resonant frequency of the cavity with respect to both the exciton binding energy and the exciting THz photon energy simply by moving the sample side to side. In this way, the incoming optical excitation encounters the cavity at a different thickness. The significance of this is discussed later. Fig. 7.4 Depiction of QW micro-cavity sample. (a) Side view of QW stack sandwiched between two DBR s and set on a substrate. (b) Top view of sample demonstrating exaggerated wedge shape and tunability of sample resonance.

124 When the weak optical excitation strikes the QW micro-cavity sample, the following reflectivity spectrum is observed. 11 Fig.7.5 Reflectivity spectrum of QW micro-cavity sample. The InGaAs QW sample actually only contains one absorption resonance in this region, however, we clearly see two sharp absorptions. These two absorption resonances are a direct observation of the exciton-polariton splitting described above. As with the bare QW sample of Chapter 6, we see very clear, sharp lines indicating a high quality sample. 7.4 Experiment Experimental Setup The experimental arrangement is shown below.

125 111 Fig. 7.6 Experimental setup used with single-cycle micro-cavity QW. The apparatus for this experiment is essentially the same as that used in chapter 6 for the bare QW experiments with a few key differences. First, DFG is not used for the generation of THz waves. Therefore, the fully compressed output from the amplifier is used to generate a strong, single-cycle THz pulse by the process of optical rectification described in chapter 3. Additionally, the compressor is not needed since the probe is already compressed. Another practical difference with this experiment is that data is gathered in a reflective geometry instead of a transmission through the sample. The reflection scheme caused some practical problems as to how to intercept the reflected optical probe beam without interrupting the input. The result was a careful construction of a sample mount that allowed us to finely tune the position of the sample with five degrees of freedom. These are shown in the figure below.

126 11 Fig. 7.7 QW micro-cavity sample mount degrees of freedom. As figure 7.7 shows, the aperture in the last parabolic mirror that the reflected optical probe must pass through twice was only about 1 mm wide requiring fine control over the sample position. The last difference is the presence of a white-light generation apparatus and a band-pass filter. The white light generation setup, shown below, is necessary because the exciton resonance for the InGaAs QW s occurs at approximately 83 nm. Fig 7.8 Simple white light continuum generation setup.

127 113 The setup in figure 7.8 allows us to expand our narrow probe pulse centered at 8 nm to an extremely broad range of wavelengths, and by use of a band pass filter, pick out a range centered on 83 nm. The broad band generation, or super-continuum generation as it is sometimes called, is accomplished by focusing the extremely short pulse onto the sapphire crystal. Although this method has been used for decades for various applications, the physical process is still not completely understood, though it is widely accepted that self-focusing and various forms of self-phase-modulation are at the heart of the process [15]. Although our oscillator is easily tunable up to central frequency of 83 nm, we opted for a white light generation setup for the following reasons: (1) The THz pulse generation is more efficient at 8 nm, and as we are looking for nonlinear effects, we did not want to sacrifice THz intensity which falls off as the square of the optical pump field, and () since the probe must be weak, we could accept a loss in intensity of the probe caused by the extra optics in the probe line. Other than these few differences, the experimental details were much the same as in the bare QW experiment. Strong, single-cycle THz pulses were generated by optical rectification using 8 nm, 1 fs optical pulses. Weak, optical pulses centered at 83 nm were used to probe the system. These were coincident on the plane of the sample which was held at ~5 K with liquid helium in a cryostat. The waveforms were obtained by EO sampling in a 1 mm thick ZnTe crystal. Additionally, we measured the incident THz power using a silicon bolometer. The THz field reached approximately 1 kv/cm. Spectral modulations were measured by observing the reflectivity, R(ν), of the optical

128 probe with a spectrometer. By changing the delay between the THz pulse, and the optical probe, the time evolution of R(ν) is mapped out Experimental Considerations With the above background and setup, we can more carefully describe the complexity of the experiment performed. To the best of our knowledge, no prior THz studies of this kind have been done on optical micro-cavities. As such, there was some question as to how the THz radiation would interact with the micro-cavity QW. There are many relevant energies present in the experiment, all of which correspond to THz photon energies: (1) As in the bare QW experiment, the exciton 1s-p transition falls in the THz photon range, () The LEP/HEP-p transition energies are in the THz photon range, and (3) The splitting of the HEP and LEP resonances also falls into this range. It was not known to what degree the THz photon might couple the HEP and LEP states. A semi-classical analysis predicted a huge enhancement of the nonlinear polariton responses when THz radiation is tuned to the polariton splitting [151], which, however, contradicts the quantum symmetry of the polariton wave functions, i.e., dipole transitions between the two polariton modes are forbidden. Additionally, it was not known whether one of the two following contradictory situations would prevail. In the first scenario, THz photons couple the excitonic 1s and p states as in our previous works [118, 136], and these effects are enhanced by the microcavity. In the second scenario, the cavity mode couples to the exciton first forming polariton states, and these states are in turn

129 coupled to the excitonic p state by the THz photon. For clarity, these two situations are depicted below in figure Fig. 7.9 Possible quasi-particle dynamics in micro-cavity QW. (a) THz photon only couples to exciton, and the cavity mode couples to the Rabi-split levels. (b) THz photon couples to polariton λ-system and can split either LEP of HEP level. The particular ordering of when the THz photon couples to which quasi-particle is not important so much as answering the question of whether the THz couples to the exciton alone or the polariton. This question was answered in other work performed in our lab with narrowband THz pulses produced by the DFG [15]. By tuning the photon to particular transitions, we found that the THz couples to the polariton states, and that the correct dynamics of the LEP, HEP, and p states are those of the lambda system depicted in figure 7.9(b). With this in mind, we pumped the system with a broadband, single-cycle THz pulse as well. The single-cycle pulse provided several advantages that were worth

130 116 exploring. First, the single-cycle setup is much less complicated. Because of its simplicity, and because of the use of a fully compressed optical pump, the source outputs more intense electric fields than the multi-cycle pulse is capable of producing. Also, the pulse duration of the single-cycle pulse is ~1 ps (compared to the DFG pulse duration of ~4 ps), allowing us greater resolution in mapping of the THz effects. However, the simplicity of the THz generation is accompanied by an increase in the complexity of the dynamics present in the system. The broadband nature of the pulse implies that the observed effects will no longer be dominated by the resonant excitation of a single transition. Instead, multiple frequencies can simultaneously cause resonant excitations. Additionally, these effects can be augmented or diminished by various non-resonant field induced effects. These effects could include, but are not limited to, spectral variations due to the Dynamic Franz-Keldish Effect, the A.C. Stark effect, and ponderomotive effects. Similar considerations were necessary in past experiments with single-cycle pulses [118]. Unfortunately, a full theoretical comparison is not yet available, as our experimental collaborators have not completed a treatment for this system. However, much promising and interesting experimental data has already been gathered, so we can at least examine the preliminary results from a phenomenological point of view. 7.5 Results One of the first experimental procedures we undertook was to verify the polariton dispersion in the absence of a THz pulse. To verify the dispersion, we pumped the cooled sample with the weak optical probe, and scanned the sample along the Y-axis as

131 117 defined by figure 7.7. As we scan along the wedge-shaped sample, the cavity resonant frequency changes with respect to the exciton binding energy. This energy difference is what will be referred to as the cavity detuning,. δ c = is the point of c c 1s minimum energy difference between the LEP and HEP modes, and corresponds to the point in figure 7.5 where the uncoupled photon and exciton dispersion relations cross. Figure 7.1 shows the result of this scan. Fig. 7.1 Microcavity QW detuning measurement. (a) Scan of the cavity detuning, δ c = ν c ν 1s. (b) Individual absorption spectra with white lines a guide. These figures show the clear splitting of the single exciton absorption into two polariton modes. As the cavity resonance is changed, the polariton dispersion relation emerges. Characterizing the detuning is important because the polariton binding energy determines the LEP/HEP-p transition energies. The simplest way to model this dispersion relation is with a coupled oscillator model. Based on this simple model we can approximate the polariton energies as a function of the detuning by the following equations

132 118 E E HEP LEP c c c c w w (7.4.1) In equation 7.4.1, δ c is the cavity detuning, and w is the mode separation at δ c =. The plot below shows a plot of the position of the LEP/HEP resonances with respect to E 1s versus the detuning. Fig Peak positions of the LEP and HEP modes vs. cavity detuning. The 1s-exciton resonance is set to the origin. The measurements of the THz induced nonlinear effects were made at δ c = -.3, -1.9, -1.7, +.3, +1.3, and +1.7 mev. As we see from figure 7.11, this coupled oscillator model fits the data very well. As a result, we were able to create a table of wavelength versus detuning values from equation

133 7.4.1 in order to identify a particular detuning rapidly during experimentation simply by observing the absorption line wavelengths and matching them to the table. 119 The next experiments performed were the time-resolved measurements described in section At several different detunings, we scanned the delay between the optical probe and THz pump in order to map out the THz interaction with the polariton modes. The results of this time-resolved study were several extremely interesting observable nonlinear effects. The figure below shows a single-cycle THz pulse used in this experiment and its corresponding spectrum. Fig. 7.1 Single-Cycle THz pulse used in time-resolved study of QW micro-cavity. Inset shows the pulse spectrum.

134 1 Scanning this pulse with respect to the optical probe revealed the following time evolution of the THz interaction with the polariton states. Fig Time-resolved differential reflectivity. (a) Birds-eye view of differential reflectivity (b) Lighting effects applied to better illustrate interference effects. From figure 7.13 we notice several strong effects. First we notice clear nonlinear effects changing the optical properties of the polariton absorptions around zero time delay. These effects are indicated by the white portions surrounded by darker areas in figure 7.13(a). These effects are caused by broadening of the absorption, and decreased absorption at the center frequencies of the polariton mode as the THz couples the polariton states to the p exciton state. One other very pronounced effect is the development of the spectral oscillations at negative time delays. Since the single cycle THz pulse is broadband, these oscillations are correspondingly spread out further around the polariton resonances. As mentioned in section 6.4.1, this is a hallmark of coherent

135 11 polarization. The coherent LEP and HEP polarizations are driven by the optical pulse which arrives first. As these polarizations decohere, the frequency of these polaritonic states spreads around the resonant frequency. This spreading is the cause of the characteristic oscillatory fanning out at negative time delays. In this experiment, because the THz field is very strong, and the single cycle pulse is so short, we have an unusually high resolution. We can see a clear interference of the polariton states as they decohere in the cross-hatched pattern developed between the LEP and HEP states. This pattern can be seen in both parts of figure A lighting effect was added to figure 7.13(b) in order to more clearly demonstrate the interference by bringing out shadows cast by maxima over the minima. The result is an amazing physical demonstration of the time development of a purely quantum mechanical interference effect. As mentioned above, one of the distinct advantages of using single-cycle THz pulses is the corresponding increase in THz intensity, which is demonstrated in impressive fashion by another preliminary result shown below.

136 1 Figure 7.14 Time resolved reflectivity of the microcavity sample. This scan was done for a relatively long scan (~3 ps). The plot shows the time evolution of the reflectivity of the microcavity sample over a period of time long enough to capture the effects of successive THz pulses caused by internal reflections of the optical pulse inside the ZnTe generation crystal. These reflections are commonplace, but are seldom useful since the amplitude of each subsequent pulse falls off sharply with respect to the prior pulse. As we are usually concerned with the maximum THz pulse energy to excite nonlinear effects, we often only closely examine the effects of the first pulse. However, in this case, we can see that the effects due to the following pulses are hardly discernable from those of the first pulse. These strong modulations indicate that the nonlinear effects are saturated even at the relatively lower field strengths of the weaker THz pulses. The large modulations also