Construction of Time-And-Angle-Resolved

Construction of Time-and-Angle-Resolved

Photoelectron Spectroscopy on Correlated Materials

Shuolong Yang

June 9, 2010

Abstract Angle-Resolved Photoelectron Spectroscopy (ARPES) has been a major experimental technique in studying the electronic band structure of condensed matter systems. The traditional ARPES system resolves binding energies and in-plane electron momenta of electronic systems. It is of great scientiﬁc interest to further resolve the information of electron dynamics. In this thesis, I give a brief review of the theoretical and experimental backgrounds of the Femtosecond Time-Resolved ARPES technique. I report the construction of such a system, which consists of an ultrafast 1.5 eV pump beam and 6 eV probe beam, and a Scienta SES200 electron analyzer. An optical au- tocorrelator veriﬁes that the duration of the pump laser pulse is 75 fs; An ARPES experiment on an Sb sample demonstrates that the angular resolution of the electron analyzer is less than 0.34◦ and that the energy resolution is less than 50 meV.

1 Contents

1 Acknowledgments5

2 Theoretical Background6 2.1 Introduction to Ultrafast Electron Dynamics...... 6 2.1.1 Motivation to Study the Time Evolution of Correlated Materials...6 2.1.2 Time Scales of Diﬀerent Types of Interactions in Correlated Systems7 2.2 Basic Principles of Ultrafast Laser Setup for Pump-probe Technique.....9 2.2.1 Mode-Locking Scheme for Ultrafast Laser Pulse Generation...... 9 2.2.2 Group Velocity Dispersion...... 11 2.2.3 Higher Harmonic Generation...... 12 2.3 ARPES Basics...... 14 2.3.1 ARPES as a Measure of One-electron Spectral Function...... 14 2.3.2 ARPES Instrumentation and Resolution Concerns...... 16 2.3.3 Pump-probe Technique...... 19

3 Construction and Characterization of the Ultrafast Laser Output 20 3.1 Complete Ultrafast Laser Setup...... 20 3.2 Optical Autocorrelation System and Characterization of the Time Profile.. 22 3.2.1 Introduction to Optical Autocorrelation System...... 22 3.2.2 Characterization of the Time Profile of the Ultrafast Laser Pulse... 23 3.3 Spectral Profile of the Probe Beam...... 24

4 Calibration and Characterization of the Electron Analyzer 25 4.1 Principles of the Angular Test Device...... 25 4.2 Calibration Results and Analysis...... 27

5 Testing ARPES Experiment on Sb 29 5.1 Introduction to Rashba Splitting...... 29

2 5.2 Measurement of the System’s Performance...... 31

6 Further Calibration and Optimization 34

7 Conclusion 35

3 List of Figures

1 Time and energy scales for diﬀerent interactions in correlated materials...... 8

2 a) Ultrafast Gaussian beam proﬁle in the frequency domain; b) ultrafast Gaussian

beam proﬁle in the time domain...... 10

3 Physical picture of photoemission process: a) excited photoelectrons; b) diagram of

energy conservation in photoemission[11]...... 14

4 a) Working scheme of hemispherical electron analyzer; b) typical ARPES spectrum[19]. 17

5 Pump-probe technique for Time-Resolved ARPES measurement...... 19

6 Complete optics setup for pump-probe measurement...... 20

7 Flip mirror setup for ﬁnding spatial and temporal overlap...... 21

8 Optical autocorrelation setup...... 22

9 Autocorrelation results for the Ti-Sapphire output: a) eﬀect of time-bandwidth

duality, 17 nm vs 21 nm; b) eﬀect of chirp mirror compensation...... 23

10 Spectral proﬁle of the probe beam...... 25

11 Angular test device schematics...... 26

12 ARPES spectrum of the angular test device ...... 27

13 a) Momentum Distribution Curve and ﬁtting; b) Angle-integrated Energy Distri-

bution Curve and exponential ﬁtting...... 28

−1 14 Sb band structure at kx = 0.035A˚ : dotted bands represent the split surface states 31

15 ARPES measurement on Sb(111) by our constructed system:a) measured band

structure; b) EDC plots ...... 32

16 Fermi surface mapping of Sb(111): a) Fermi surface mapping using our constructed

system; b) comparison with the published data by Chen et al[4]...... 33

4 1 Acknowledgments

This work has received tremendous help from Prof. Zhi-Xun Shen’s research group, De- partment of Physics at Stanford University and Stanford Vice Provost for Undergraduate Education. Special thanks go to Patrick Kirchmann and Jonathan Sobota, whom I have been closely collaborating with and learning from. Photoemission technology has been de- veloping vastly for the past two decades. It took the effort of several generations of scientists and engineers to implement this technology and produce significant scientific data. It was the selfless help from my colleagues that made it possible for me to master the fundamentals and furthermore contribute to this field in only one year. I also attribute this work to the people who have greatly helped me with study and life as an international undergraduate student, among whom are Prof. Douglas Osheroff, Prof. David Goldhaber-Gordon, Prof. Hari Manoharan, Prof. Daniel Bump, Mr. Rick Pam and Ms. Elva Carbajal.

5 2 Theoretical Background

Time-Resolved ARPES is a powerful tool in studying the ultrafast electron dynamics in correlated systems. In this section, I will brieﬂy explain how the underlying microscopic interactions and elementary excitations in a complex correlated system can be disentangled in the time-domain due to their distinctively diﬀerent time scales. It is this interest that drives us to construct a Time-Resolved ARPES system. In the next two sub-sections, I will introduce the basic principles of ultrafast laser and ARPES technologies.

2.1 Introduction to Ultrafast Electron Dynamics

2.1.1 Motivation to Study the Time Evolution of Correlated Materials

Quasi-particle excitations exchange energies and momenta through scattering processes, which are deﬁned by the underlying microscopic interactions. One way of studying these interactions is to excite the electronic system with an ultrafast pump laser pulse in the femtosecond regime and record by a second time-delayed probe laser pulse, how the electronic band structure evolves as a function of pump-probe delay. In particular, we use a femtosecond infrared pump beam to excite the system and a ultraviolet probe beam that photo-emits the electrons. One of the central quantities in such an experiment is the population decay rate Γ of electrons excited above the Fermi level EF . Several scattering schemes contribute to Γ[13]:

1e Γ = Γe−e + Γ + Γe−ph + Γe−def (1)

1e Here Γe−e,Γ ,Γe−ph and Γe−def denote the decay rates of inelastic electron-electron, elastic electron-electron, electron-phonon and electron-defect scatterings, respectively. By studying the decay rate of the excited electron population, which are termed hot electrons since they are not in equilibrium with the lattice, it is possible to retrieve the information about diﬀerent scattering mechanisms. In particular, the time evolution of hot electrons

6 in metallic materials has been investigated in great detail[6][9]. In general, Γe−e dominates

Equation1 for large excitation energies E − EF kBT ;Γe−ph dominates for low excitation

energies E − EF kBT . It should be noted that the preceding discussion of interaction disentanglement only concerns metallic materials, for which the Fermi Liquid Theorem is widely applicable. The situation is different for highly-correlated materials such as Mott Insulators or High Tc Superconductors, for which the failure of Fermi Liquid Theorem has been demonstrated by ARPES experiments[7]. The time evolution of electronic states in correlated systems has emerged as a major research field in that it provides a new perspective to evaluate current theoretical models. In particular, as in the example of high Tc superconductivity, the small isotope shift effect in optimally doped samples[1] and the d-wave symmetry of the superconducting gap[7] evidence an argument against the mechanism of phonon-mediated pairing. Dynamical information of electron-phonon interactions shines light on evaluating this type of arguments.

2.1.2 Time Scales of Diﬀerent Types of Interactions in Correlated Systems

The formalism of characterizing the different types of interactions in a electronic system remains valid in the context of correlated materials. In order to reveal the physical picture of ultrafast electron dynamics in correlated materials, it is crucial to survey the time scales of different types of interactions, as illustrated by Figure1[17]. It is evident from Figure1 that different types of interactions generally have different time scales. In particular, the electron-electron thermalization process has a characteristic time varying from a few femtoseconds to hundreds of femtoseconds, and generally happens much faster than the electron-phonon coupling and the coherent phonon interactions. In this regard, if we pump a correlated system using photon absorptions, electrons will be thermal- ized first to a high temperature. These hot electrons reach the peak temperature at the time scale of 100 fs. The electron temperature starts decreasing by means of inelastic electron-

7 Figure 1: Time and energy scales for diﬀerent interactions in correlated materials. electron scattering and electron-phonon interactions. Some lattice modes are coupled with the hot electrons and thus become hot phonons. At this stage, there exist three distinct

temperatures in the correlated system: electron temperature Te, hot phonons’ temperature

Tph−hot and uncoupled phonons’ temperature Tph−un. Tph−hot increases while Te decreases until at several ps they are in equilibrium. The remaining two distinct temperatures continue converging for several ns before the whole system is eventually thermally equilibrated. The above physical picture is mathematically depicted by the following decay equations[16].

∂T 3λΩ3 n − n P e = − 0 e ph−hot + (2) ∂τ hπk¯ B Te Ce

∂T C 3λΩ3 n − n T − T ph−hot = e 0 e ph−hot − ph−hot ph−un (3) ∂τ Cph−hot hπk¯ B Te τβ

∂T C T − T ph−un = ph−hot ph−hot ph−un (4) ∂τ Cph−un τβ

In the above equations, Ω0, P and τβ are the phonon energy, the external heat transfer

and the characteristic time of phonons’ anharmonic decay, respectively; ni’s and Ci’s are the

8 2 respective population densities and specific heats; λΩ0 is a measure of the electron-phonon coupling intensity. By fitting the data for cuprate superconductor compound Bi2212, Perfetti et al. concluded that only about 20% of the phonons are coupled with electrons in this process and argued against the mechanism of purely phonon-mediated pairing[16]. The above method is effective yet still unable to disclose the full image of ultrafast electron dynamics. Firstly, the momentum dependence of population evolution awaits investigation for cuprate superconductors such as Bi2212. Secondly, little evidence has been found for that the superconducting state is preserved when pumped with 30 kHz repetition rate, 100 µJ/cm2 fluence beam[16]. Thirdly, there are evidences that the multi-temperature model fails to explain the non-equilibrium asymmetry in Falicov-Kimball model[15]. It is for these reasons that we were motivated to build a Time-Resolved ARPES system that will enable us to investigate ultrafast electron dynamics with sub-100-fs time resolution, sub-50-meV energy resolution and sub-0.5-deg angular resolution. The discussion of how to achieve and verify these instrumental advances will be the main topics for the rest of this thesis.

2.2 Basic Principles of Ultrafast Laser Setup for Pump-probe Technique

One of the most important and challenging tasks in a Time-Resolved ARPES setup is to make ultrafast laser pulse, since it determines the time resolution of the system. This involves achieving the mode-locking scheme, compensating the broadening eﬀect induced by the Group Velocity Dispersion and establishing higher harmonic generation systems.

2.2.1 Mode-Locking Scheme for Ultrafast Laser Pulse Generation

It is well-known that the monochromaticity of single-mode laser beams gives rise to a wide spread in the time domain due to the fundamental limit of time-bandwidth product. In order to have short pulses in the time domain, however, one has to coherently couple a large number of modes such that these modes have time-independent phase diﬀerence. In this scheme, the laser beam is said to be mode-locked[18]. This physical picture is illustrated in

9 Figure 2: a) Ultrafast Gaussian beam proﬁle in the frequency domain; b) ultrafast Gaussian beam proﬁle in the time domain.

Figure2. As illustrated in Figure2, suppose that there is a mode-locked laser beam E(t) =

P∞ 2n∆ω 2 En exp iωnt, where ωn = ω0 + n∆ω and En = E0 exp [−( ) ln 2]. Here ω0, ∆ω −∞ ∆ω0 and ∆ω0 are the central frequency, the frequency spacing between adjacent modes and the Full Width Half Maximum (FWHM) of the beam envelope, respectively. It is easily shown that by Fourier transform, the laser intensity in the time domain is proportional to exp [−( 2t )2 ln 2]. Hence we have the fundamental limit of time-bandwidth product as in ∆T0 the following expression.

√ ∆T0∆ω0 = 2 2 ln 2/π (5)

Although Equation5 only applies to Gaussian waveform, the time-bandwidth product is always bounded from below by a unity-scale quantity. The key to ultrafast laser pulse generation is to find a medium with a broad gain profile that covers a large number of phase-locked modes. Several techniques have been used for this purpose such as amplitude modulation, frequency modulation and saturable absorber-amplifier method[18]. In our case, we use Ti-Sapphire laser that makes use of the intensity-differentiated self-focusing mode-

10 locking scheme. It is also known as self-locking. The Kerr effect in the lasing medium behaves like a converging lens and cuts the leading edge of the time profile while enhancing the peak value. The beam is then significantly shortened to fs-scale in the time domain. It should be noted that people have pushed the ultrafast pulse generation far beyond sub- 100-fs-scale. In 2007, Cavalieri et al. reported the generation of sub-4-fs near infrared output which covers only 1.5 cycle in one pulse[2]. In fact, in 2005 Lindner et al. have already reported an attosecond experiment that shines new light on quantum computing[14]. It is still highly non-trivial to fully facilitate attosecond spectroscopy. Yet, it is likely that the technological advance will bring new perspectives to the fundamental research in physics. However, in a Time-Resolved ARPES experiment we have to optimize the time-bandwidth product to still detect discernible peaks in the electron spectra. This results in time-resolutions of 10-100fs as ideal probes for Time-Resolved ARPES experiments.

2.2.2 Group Velocity Dispersion

Ultrafast laser pulses contain a broad range of frequency components. If wave vector values for diﬀerent frequencies are not uniform, the spatial waveform will be time-dependent. If we

0 approximate the wave vector to the second order term, namely k(ω) = k(ω0) + k (ω − ω0) +

1 00 2 2 k (ω − ω0) , a Gaussian frequency proﬁle will be Fourier-transformed into the following form[18].

r Γ(x) x x 2 (t, x) = exp [iω0(t − )] exp [−Γ(x)(t − ) ] (6) π vφ(ω0) vg(ω0)

where

ω dω 1 1 v (ω ) = ( ) , v (ω ) = ( ) , = + 2ik00x (7) φ 0 k ω0 g 0 dk ω0 Γ(x) Γ

Γ is proportional to the square of the frequency Gaussian width; vφ and vg are phase and group velocities, respectively. It is easily seen from Equation6 that the phase of the

11 electromagnetic wave is transmitted with velocity vφ, and the peak value in the time proﬁle

00 d2k d 1 is transmitted with velocity vg. k = ( )ω = ( )ω is called the Group Velocity dω2 0 dω vg(ω) 0 Dispersion (GVD). Deﬁne ξ = 2Γk00. The second exponential part in Equation6 can be rewritten in the following form[18].

Γ x 2 ξx x 2 exp [− 2 2 (t − ) + i 2 2 (t − ) ] (8) 1 + ξ x vg 1 + ξ x vg

The real part gives the wave intensity information, and clearly a broadening eﬀect follows

1 from the function 1+ξ2x2 . This duration broadening occurs commonly when ultrafast pulses go through dispersive elements, such as lenses and nonlinear crystals (Section 2.2.3). It is a major concern when we conduct ultrafast experiments. Usually people use devices such as prism compressors or chirp mirror compressors to compensate the GVD losses. These techniques will be demonstrated and veriﬁed in later sections.

2.2.3 Higher Harmonic Generation

In a pump-probe process, the pump beam ﬁrst pumps the sample with photon energy less than the sample’s work function. Photocurrent does not occur, but hot electrons and hot phonons are generated. The probe beam arrives at a time delay ∆τ, and probes the system using photoemission. The essential technique to obtain our 6 eV probe beam is Higher Harmonic Generation (HHG) process. It utilizes the second order terms in some nonlinear crystal’s susceptibility. This process also has to satisfy the conservation laws of energy and momentum in order to generate a macroscopic higher order radiation.

k(ω1 + ω2) = k(ω1) + k(ω2) (9)

Note that the energy conservation is already satisfied by summing the two frequencies to obtain the new frequency. In order to maximize the effective interaction zone, we operate in the collinear configuration, and Equation9 becomes scalar addition. Because c/n = ω/k

12 where n is the refractive index, Equation9 turns into n3ω3 = n1ω1 + n2ω2. The condition for frequency addition is eventually n3 = n1 = n2. It therefore suffices to finding an incident angle at which the refractive index for the output beam matches that for the input beam. This angle is known as the phase matching angle[8]. In order to obtain the 6 eV probe beam from the 1.5 eV fundamental beam, we apply the two-stage Second Harmonic Generation (SHG) system. The fourth harmonic output from the second stage will be the probe beam. The two most important parameters in an SHG process is efficiency and operating wavelength range. People have been able to find and characterize numerous kinds of functional frequency doubling or trippling crystals, such as Potassium Dihydrogen Phosphate (KDP), Lithium Triborate (LBO), Beta-Barium Borate (BBO), etc. The SHG efficiency is generally higher than 10% with appropriately focused input beams. This efficiency can be further improved with thicker crystals, which, however, induces larger GVD losses and elongates the laser pulse in the time domain. As for the issue of wavelength range, most nonlinear crystals have the Type-I SHG cutoff wavelength higher than 400 nm[10]. KBe2BO3F2 (KBBF) crystal has an exceptionally low cutoff wavelength, and has been demonstrated to work with 358.7 nm second harmonic to generate the fourth[3]. However, practical concerns such as low damaging threshold limits further applications of KBBF crystals. Not all frequency doubling process occurs at collinear conditions. In fact, an important apparatus which we built for characterizing the beam profile, and which will be discussed in greater detail in Section 3.2, is an optical autocorrelation system. It mixes two non- collinear beams to generate the second harmonic output. This process occurs when spatial and temporal overlaps are both satisfied and hence provides an approach to measure the time duration of one pulse.

13 Figure 3: Physical picture of photoemission process: a) excited photoelectrons; b) diagram of energy conservation in photoemission[11].

2.3 ARPES Basics

Time-Resolved ARPES inherits every feature of the traditional ARPES systems with an extra functionality of resolving time information. It is essential to understand the underlying principles of the traditional ARPES systems before moving onwards to the construction of the Time-Resolved ARPES system.

2.3.1 ARPES as a Measure of One-electron Spectral Function

The phenomenon of photoemission was discovered by Hertz[11] in 1887. It was not until 1905 that Einstein gave a theoretical explanation to the experiment by invoking the quantum nature of light. When photons are injected to an electronic system, electrons interact with the photons and the energy is transferred from photons to electrons with a certain probability. If the photon energy is larger than the material’s work function, excited electrons can be emitted to the free space, resulting in a photocurrent. This picture is demonstrated in Figure3.

14 Figure3a) is a diagram illustrating the process of photoemission. Note that in this conﬁguration, the horizontal momentum is conserved while the vertical momentum is not due to the surface anisotropy. Figure3b) demonstrates the energy conservation in photoemission processes. The basic conservation laws lead to:

Ekin = hν − φ − |EB| (10)

p |pk| =h ¯|kk| = 2mEkin sin θ (11)

It is evident from Equation 10 and 11 that by measuring the kinetic energy Ekin and the momentum parallel to the sample surface pk, we can directly measure the binding energy EB and the parallel momentum kk of a particular electronic state. Note that here we ignore the photon momentum in Equation 11. This is an appropriate approximation given low photon energy. In our case, the probe photon energy will be 6 eV, resulting in a photon momentum of 0.003 A˚−1. This is less than 0.2 % of the typical Brillouin-zone size of cuprates[7]. We employ the sudden approximation, which treats the emission of each electron as a sudden process with no post-collisional interactions with the rest of the system. Then the photocurrent I(k,Ekin) can be expressed in the following relation[7].

I(k, ω) ∝ I0(k, ν, A)f(ω)A(k, ω) (12)

Here k = kk; ω is the electron energy with respect to the Fermi level; I0(k, ν, A) is proportional to the squared one-electron matrix element; f(ω) is the Fermi-Dirac distribution function; A(k, ω) is precisely the one-electron spectral function. Hence, ARPES enables us to directly reveal the E-k band structure. The past two decades have seen a signiﬁcant development in ARPES instrumentation that facilitates several key discoveries in condensed matter physics, among which is the study of cuprate superconductors. Inexplicable within the well-known BCS frame, cuprate

15 materials triggered a tremendous amount of scientiﬁc discussion about new mechanisms of superconducting states. It should be noted that the technological advances in ARPES instrumentation historically nurtured many discoveries that can only be made at extremely high resolution. The bilayer splitting of the in-plane electronic structure in Bi2212 used to be unresolved and thus trigger intense debate[7]. The controversy of the Fermi surface topology of Bi2212 was not resolved until Chuang et al. observed the bilayer splitting with the angular resolution less than 0.4◦ and the energy resolution less than 20 meV[5].

2.3.2 ARPES Instrumentation and Resolution Concerns

In spirit of the significant discoveries reviewed in Section 2.3.1, it is essential for us to build our instrumentation with optimal resolutions. This section presents an analysis of the important factors that determine ARPES resolutions. We proceed by first reviewing the Ultra High Vacuum (UHV) condition, under which most ARPES experiments are conducted. ARPES is a surface sensitive technique since the mean free path of the photo-emitted electrons amounts only to a few atomic layers. UHV condition is thus the first requirement to guarantee high resolutions. The general pressure requirement is 5 × 10−11 Torr[7]. This is because the time to build up a monolayer contamination on sample surfaces is about 10 hours at 10−10 Torr but only a few seconds at 10−6 Torr[13]. To realize this UHV condition, we use various types of pumps, such as turbo pumps, ion pumps, cryo pumps, non-evaporative getter pumps, etc. A combination of these pumps is usually used to pump the main chamber, in which ARPES experiments are conducted. Figure4a) is a cartoon illustrating the working scheme of a typical hemispherical electron analyzer. By travelling through the electron lens system, Photoelectrons are accelerated/decelerated to the neighborhood of a certain pass energy and focused to the entrance slit. A radial electrical field is applied inside the hemisphere, and the kinetic energy and 1-D momentum (x direction as shown in the diagram) are resolved on the detector. Figure4b)

16 Figure 4: a) Working scheme of hemispherical electron analyzer; b) typical ARPES spectrum[19]. demonstrates a typical ARPES spectrum. We define the size of the smallest feature that can be resolved on the energy axis as energy resolution, and the one on the angular/momentum axis as angular/momentum resolution. Note that angle and momentum are directly related by Equation 11, and are sometimes synonyms of each other in the context of ARPES experiment. In practice, we vary sample orientation with respect to the analyzer. Then we obtain another E − kx spectrum at a different ky. Systematically scan over ky, and we will map out the band structure in the 3-D space of E − kk. The energy resolution of an ARPES experiment is mainly determined by two factors: the finite bandwidth of the light source spectrum and the broadening effect induced by the analyzer itself. The former will be discussed in separate sections, whereas the latter is described by the following expression[7].

w ∆E = Epass (13) R0

17 Epass, w and R0 stand for the pass energy, the entrance slit width and the average radius

of the hemisphere, respectively. It is thus straightforward to reduce Epass and w while

increasing R0 in order to enhance the energy resolution. At the same time, besides the engineering difficulties, reducing w will result in a loss in photocurrent intensity, which will hinder the Signal-to-Noise Ratio (SNR). Scientists have to deal with this trade-off and obtain best energy resolutions by optimizing these parameters. For Scienta SES200 analyzers, the energy resolution can be made to a few meV, which makes the analyzers capable of resolving most of the correlated energy features such as superconducting gaps and pseudo gaps. In fact, for Time-Resolved ARPES experiments, we have to sacrifice the energy resolution in order to obtain a good time resolution due to the energy-time duality[15]. Hence the fundamental limit to the energy resolution is the finite bandwidth of our laser output. It will be demonstrated in Section 2.3.3. The momentum resolution can be determined directly from Equation 11 by differentiating both sides.

q 2 ∆kk = 2mEkin/h¯ cos θ∆θ (14)

Here ∆θ is the angular resolution of the electron analyzer. It is clear that we have to use lower photon energy to enhance the momentum resolution. In that sense, our 6 eV probe beam will benefit the momentum resolution compared with a synchrotron light source. In the meantime, we are again confronted with the trade-off between higher SNR and higher resolution. For the type of analyzer we used, Scienta SES200, the angular resolution can be made in principle as small as 0.2◦ for 21.2 eV synchrotron radiation[7], which is good enough in resolving fine structures such as the bilayer splitting of Bi2212 Fermi surfaces. We will demonstrate in our data analysis section that despite the fact that our analyzer was not designed to work with 6eV laser beams, it still produces decent data with sub-0.5-deg angular resolution.

18 Figure 5: Pump-probe technique for Time-Resolved ARPES measurement.

2.3.3 Pump-probe Technique

The preceding discussions of the basic principles of ARPES and ultrafast laser pulse combine to give a full description of the pump-probe technique we use for Time-Resolved ARPES experiments. Figure 2.3.3 illustrates the schematics. A 1.5 eV beam pumps the sample, and a 6 eV beam comes after a time delay of ∆τ to probe the band structure information. Since we verify that the time duration of our laser pulse is sub-100-fs, we will be able to probe the rich physics induced by electron- electron and electron-phonon interactions as explained in Section 2.1. We point out that in order to obtain a good time resolution, we have to enlarge the bandwidth of the laser spectrum and thus sacriﬁce the energy resolution[15]. This is due to the energy-time duality ∆E∆T ∼ 1. In fact, our experimental data reveals that our energy resolution is almost exclusively determined by the laser bandwidth. In the following sections, I will report our experimental analysis for the time, energy and momentum resolutions of our Time-Resolved ARPES system.

19 Figure 6: Complete optics setup for pump-probe measurement.

3 Construction and Characterization of the Ultrafast Laser Out-

put

The photon source determines or partially determines several crucial features of our system, such as time resolution, energy resolution and photocurrent intensity. We constructed and characterized our laser output using spectroscopic and temporal approaches.

3.1 Complete Ultrafast Laser Setup

Figure6 illustrates the complete ultrafast laser setup of our system. It is basically comprised of two separate beam paths: one for the pump beam and the other for the probe. The well- tuned Ti-Sapphire oscillator outputs ultrafast laser pulse with 800 nm wavelength and 80

20 Figure 7: Flip mirror setup for ﬁnding spatial and temporal overlap.

MHz repetition rate. It will be shown by the optical autocorrelation data that the oscillator output has a pulse duration about 75 fs. A beam splitter splits the beam into pump and probe branches. The one going through the Two-stage Fourth Harmonic Generation system will provide the probe beam. We use BBO crystals for the Fourth Harmonic Generation process. After each SHG stage, a prism compressor is used to compensate the GVD. The probe beam then goes through a delay stage which varies the delay time in pump-probe measurement. Not many complications are involved for the pump beam setup. It goes through a Chirp Mirror System, which also compensates the GVD loss in the beam path, and is then combined with the probe beam line to be focused onto the sample. Pump and probe beams have to be spatially and temporally overlapped. To accomplish the first goal, we set up flip mirrors before the two beams are focused onto the sample and change the beam paths to let the two beams be focused onto a pinhole at the same distance as the sample is from the flip mirrors (see Figure7). To accomplish the second goal, we have to approximately match the lengths of the two beams and then move the delay stage until we find the overlap by placing a fast photodiode at the pinhole’s place in Figure7. After we see the overlap, the delay stage is carefully scanned until hot electrons are created.

21 Figure 8: Optical autocorrelation setup.

3.2 Optical Autocorrelation System and Characterization of the Time Proﬁle

3.2.1 Introduction to Optical Autocorrelation System

Ultrafast laser pulse is so short that regular photodiodes are not capable of resolving the fs-scale time proﬁle. A shortcut method is to use a fast photodiode resolving Terahertz-scale radiation. In this project we pursued another approach called Optical Autocorrelation, which transforms the time measurement into a position measurement. A schematic diagram of this autocorrelation setup is shown in Figure8. As shown in Figure8, the incoming beam is split into two arms, which are recombined by a focusing lens. They are focused onto a BBO crystal. For spatially and temporally overlapped beams under phase-matching condition, a correlation SHG signal will be detected. Its intensity is subject to the following expression.

Z Iac(τ) ∝ I(t)I(t + τ)dt (15)

2∆L We move the delay stage as shown in Figure8 and it is evident that τ = c . Although

22 Figure 9: Autocorrelation results for the Ti-Sapphire output: a) eﬀect of time-bandwidth duality, 17 nm vs 21 nm; b) eﬀect of chirp mirror compensation.

τ is in the order of fs, ∆L, which is in the order of 10−2 mm, is easily measured with a high-precision motor controller. Hence we can determine the FWHM of the original pulse in the time domain by measuring the autocorrelation signal. For ideal Gaussian beams, we √ just have to scale the FWHM down by a factor of 2. As mentioned in Section 2.2.3, not all frequency addition processes are conducted in collinear condition. This optical autocorrelation system is such an example of non-collinear frequency addition. This process is also dictated by the energy and momentum conservation laws.

3.2.2 Characterization of the Time Proﬁle of the Ultrafast Laser Pulse

Our data is presented in Figure9. Figure9a) demonstrates the eﬀect of time-bandwidth duality. The outer blue Gaussian proﬁle is associated with the laser bandwidth of 17 nm and the inner one with 21 nm. The calculated temporal FWHM’s are 83 fs and 76 fs, respectively. The time-bandwidth products

23 (∆ν∆t) are 0.617 and 0.693, respectively. The shortening effect of the pulse time duration is evident, while the laser operating at 21 nm is slightly less ideal and further away from the Fourier limit (∆ν∆t = 0.441). Figure9b) demonstrates the effect of adding a chirped mirror system. Chirped mirrors delay the different frequency components of the broadband laser beam such that a negative GVD is introduced , which can compensate the dispersion of optical elements. The calculated FWHM’s are 83 fs and 75 fs for the cases without and with the chirped mirror system, respectively; the time-bandwidth products are 0.617 and 0.558, respectively. The GVD compensation pushes the laser beam towards its Fourier limit. From this analysis, we verify that the FWHM of our pump beam is 75 fs. As for the probe beam, although two prism compressors are used to compensate GVD losses, the FWHM is still slightly larger than that of the pump beam, given that there are more dispersive elements (lenses, crystals) along the probe beam path. From the analysis in Section 2.1.2, we know that as long as our beams are well below 100-fs, we are able to investigate the ultrafast electron dynamics in correlated materials. Our autocorrelation data asserts that our laser setup satisfies this requirement and is ready for Time-Resolved ARPES experiments.

3.3 Spectral Proﬁle of the Probe Beam

In addition, we measured the spectral profile of the probe beam using Ocean Optics 2000 spectrometer. The result is presented in Figure 10. Reading from Figure 10, we find the FWHM in the frequency domain is 1.68 nm, corresponding to a FWHM of 48 meV in energy scale. This energy spread completely dictates the energy resolution of the Time-Resovled ARPES measurement, since the Scienta SES200 analyzer itself only induces a line broadening effect about a few meV. This point will be verified in later sections on the testing experiment on Sb. Another point to be noted is that the optimal time duration can be made as small as 38 fs, indicating that there is still some room for time resolution improvement. Nevertheless,

24 Figure 10: Spectral proﬁle of the probe beam. the spectral proﬁle reinforces the conclusion that this ultrafast pump-probe laser system is satisfactory for Time-Resolved ARPES experiments.

4 Calibration and Characterization of the Electron Analyzer

Scienta SES200 is a high-precision electron analyzer. It is non-trivial to calibrate and characterize given that our equipment was not designed to work with a 6 eV laser beam. In this section, I will introduce the angular test device we use for calibration and characterization and the testing data we have obtained.

4.1 Principles of the Angular Test Device

The Angular Test Device (ATD) is the main tool we use for calibrating the electron optics of the analyzer. It emits electrons at discrete angles by thermionic emission and allows a measurement of angular resolution in this way. The schematics of this device is shown in Figure 11.

25 Figure 11: Angular test device schematics.

On the left of Figure 11 shows a drawing of the actual device. It consists of two major parts: a mask with parallel narrow slits and a tungsten filament sitting in the middle of the metal plate behind the mask. As shown on the right, the filament is heated by electrical current and thermal electrons are emitted to travel through the slits. Each slit determines a discrete angle. The angular spacing is 1◦ in the middle and 2◦ at sides. If the electron analyzer is well calibrated, the output ARPES spectrum will be discrete lines parallel to the energy axis. The main parameters to optimize in this calibration process is the lens table of the electron optics. Lens table is defined as an array of electrostatic voltages for the electron lenses. As introduced in Section 2.3.2, electrons are accelerated/decelerated to be close to pass energy when they enter the hemispherical space. The electrons precisely at the pass energy will land in the middle of the detector. It is the electron lenses that determine which electrons to be accelerated/decelerated to the pass energy, and the lens table allows the lenses to work with electrons with different original kinetic energies. The range of these kinetic energies determines how deep we can resolve below the Fermi level. The lens table also allows the

26 Figure 12: ARPES spectrum of the angular test device analyzer to work in a angular-resolving mode.

4.2 Calibration Results and Analysis

After optimization, our result using ATD is shown in Figure 12. There are several important observations by reading the spectrum. Firstly, we have an energy window > 1 eV and an angular window of 16◦. The latter meets the Scienta specification of angular window: 16◦. Secondly, each angle is well resolved and line width < 0.5◦. We will demonstrate that the angular resolution is actually 0.23◦ by doing curve fitting on the Momentum Distribution Curve (MDC), as shown in Figure 13a). Thirdly, the central lines are parallel and straight while there is a certain warping effect about the edge lines. This issue will not affect the general band structure topology but will deform the spectrum slightly. It can also be rectified by using this characterization spectrum and specific image processing techniques. Last but certainly not the least, the spectrum intensity is not uniform for all energies. The intensity diminishes with increasing kinetic energy due to Maxwell-Boltzman distribution. We investigate this classical picture by fitting exponential functions to the Energy Distribution Curve (EDC), as shown in Figure 13b).

27 Figure 13: a) Momentum Distribution Curve and ﬁtting; b) Angle-integrated Energy Distribution Curve and exponential ﬁtting.

In Figure 13a), we take a intensity-angle cut at Ekin = 1.37 eV. As shown in Figure 11, the intrinsic line width of ATD is 0.2◦. We approximate it as a rectangular distribution. The fitting curve we use for MDC is obtained by taking the convolution of a Gaussian-shape resolution function and the rectangular peak-shape distribution.Vary the Gaussian width of the resolution function to minimize the χ2 difference between the real MDC and the fitting curve. We obtain that the angular resolution of our analyzer is approximately 0.23◦. In Figure 13b), we integrate from −8◦ to 8◦ and fit an exponential function to the energy distribution between 1 and 2 eV. From Maxwell-Boltzman distribution, particles in a quasi- equilibrium are subject to an exponential population distribution.

n(E ) E − E 2 = exp (− 2 1 ) (16) n(E1) kBT

3 Our ﬁtting shows that kBT = 0.282eV → T = 3.27 × 10 K. Given that the melting point of Tungsten is 3695 K, this is a reasonable approximate of the true temperature of

28 the ﬁlament. Aside by the issue of how accurate this measurement is, however, our foremost motivation of showing this data is to demonstrate that our electron analyzer generates an intensity plot that truly reﬂects the population distribution of electrons. Combined with the direct observations from Figure 12, we conclude that our electron analyzer is satisfactory for doing ARPES experiments.

5 Testing ARPES Experiment on Sb

As we have demonstrated in Section3 and4, our laser output has the appropriate temporal and spectral features, and our electron analyzer is working fine with the specified angular resolution. A complete ARPES experiment is needed to verify the expected energy and angular resolutions. We chose Sb(111) because: 1) it has strong Rashba splitting effect induced by spin-orbit coupling, which can be used to characterize the system’s energy and angular resolutions; 2) it has been intensively studied by the ARPES community[20][4], making it straightforward to test our system’s performance by comparison; 3) it is fairly easy to cleave Sb samples.

5.1 Introduction to Rashba Splitting

Modern condensed matter theory formulates the idea of symmetry breaking. Phase transi- tions often happen when a certain symmetry breaks. For example, there are experimental evidences that the pseudo gap of underdoped cuprate superconductors is due to the spontaneous breaking of time-reversal symmetry[12]. I will demonstrate that the Rashba splitting stems from the breaking of space-inversion symmetry. Firstly, time-reversal symmetry leads to the following expression.

E(k, ↑) = E(−k, ↓) (17)

Space-inversion symmetry leads to:

29 E(k, ↑) = E(−k, ↑) (18)

We consider the region where both symmetries hold, such as Sb bulk states. Combine Equation 17 and 18 and we have:

E(k, ↑) = E(k, ↓) (19)

Therefore, the bulk states are spin-degenerate. For the surface states, however, the space-inversion symmetry breaks. Surface states are thus not spin-degenerate. Diﬀerent spin orientations lead to split surface bands. This phenomenon is called Rashba Splitting. This splitting eﬀect can be described using spin-orbit coupling formalism.

µ H = B (v × E).σ (20) S.O. 2c2

Here µB, c, E, v and σ denote Bohr magneton, speed of light, electrical ﬁeld, electron velocity and spin, respectively. Due to the anisotropy on the surface, E is pointing per- pendicular to the surface, v is constrained to be 2-D. Spin up or down clearly induces a diﬀerence in energy eigenvalues. Figure 14 shows published ARPES data on Sb(111) by Sugawara et al[20]. From Figure 14 we know that the momentum spacing between the two surface bands is approximately 0.07A˚−1, and that the energy spacing is approximately 0.1 eV. To resolve the two bands on both momentum and energy axes, momentum resolution is at most 0.035A˚−1 and energy resolution is at most 50 meV. These values set up the metric by which we can measure the performance of our system and verify the conclusions we draw in Section3 and4.

30 −1 Figure 14: Sb band structure at kx = 0.035A˚ : dotted bands represent the split surface states

5.2 Measurement of the System’s Performance

We conducted our measurement on Sb(111) at room temperature and low 10−10-Torr-scale pressure. Figure 15a) shows a measured band structure; Figure 15b) presents the Energy Distribution Curves (EDC) at each momentum. Note that according to Equation 11, lower kinetic energy results in larger angular spread of the spectrum. The synchrotron light source that Chen et al. used and the Helium lamp that Sugawara et al. used both have much higher photon energy than our laser output. Hence Figure 15 does not capture the full band structure. Nonetheless, it is evident from Figure 15 that our system resolves Rashba splitting on both energy and momentum axes. Note that the brighter band on the left is a bulk band; the two surface bands are resolved on the right. It follows from the ending argument of Section 5.1 that our energy and momentum resolutions are 50 meV and 0.035A˚−1 in the worst-case scenario. Recall from Section 3.3 that the bandwidth of the probe beam itself is

31 Figure 15: ARPES measurement on Sb(111) by our constructed system:a) measured band structure; b) EDC plots

48 meV. Considering that our experiment is conducted in room temperature, the thermal √ broadening is 26 meV. Combining the uncorrelated uncertainties gives us 482 + 262 = 54 meV, which is corresponding well to the energy resolution requirement. In fact, in low- temperature experiments, this resolution can be certainly improved to sub-50-meV range. The laser bandwidth, as we conclude in Section 3.3, dictates the energy resolution. As for the momentum resolution, using Equation 14 we can calculate the angular resolution to be approximately √ ¯h∆k (θ ≈ 0 and consider the splitting at the Fermi level where 2m(¯hω−φ) ◦ EB ≈ 0). We obtain that the angular resolution is at worst 0.34 . This is consistent with our characterization result in Section 4.2 that the angular resolution is about 0.23◦. Up to this moment, we have systematically veriﬁed every important feature of this Time-Resolved ARPES setup. We also plot the Fermi surface map and compare it with the established result by Chen et al[4]. As is discussed earlier in this section, the angular spread of our spectrum is larger than

32 Figure 16: Fermi surface mapping of Sb(111): a) Fermi surface mapping using our constructed system; b) comparison with the published data by Chen et al[4].

33 those obtained from synchrotron light sources. Hence with a fixed angular window, our Fermi surface map captures a smaller portion of the Brillouin zone. Figure 16a) shows this Fermi surface map using our laser setup. At the same time, it precisely matches a portion of the Fermi surface map obtained from synchrotron light source. This is shown in Figure 16b). At first glance, our Fermi surface map is vague and the features are not clearly resolved. There are several factors to consider. Firstly, we conducted our experiment at room temperature, whereas the published data was obtained at sub-10-K scale. The thermal smearing effect plays an important role. Secondly, this could result from the non-uniform value of the matrix element as in Equation 12. Thirdly, the chamber pressure does not satisfy the UHV requirement of 5 × 10−11 Torr as specified in Section 2.3.2. Lastly, the broadening can be caused partially by the space charging effect due to the strong laser power we have. This will be explained in more detail in the next section. Despite these two factors, the resolution is in fact decent if we compare our Fermi surface map with the synchrotron data in Figure 16b).

6 Further Calibration and Optimization

Time-Resolved ARPES is a high-precision experimental technique. Like any functional ARPES system, it has to be constantly calibrated and optimized to ensure high-level performance. While the major functionalities are convincingly realized with our system, there is still room for improvement with the laser setup, the UHV chamber and the electron analyzer. We can read the probe beam power from Figure 10 as being 450µW . This power already exceeds the threshold for space charging effect[13]. This effect takes place when the beam power exceeds a certain threshold, and too many photoelectrons are simultaneously generated and confined in a small space. The repulsion between electrons gives rise to the spectral line broadening. For further optimization, we have to experimentally find the power threshold and reduce the beam power to the safe range. The pulse duration of 75 fs is satisfactory to investigate electron-electron, electron-phonon interactions in most correlated materials, while it is certainly not optimized. Section 3.2.2

34 shows that the best time-bandwidth product we have obtained is 0.558, whereas the Fourier limit is 0.441. We will have to optimize the Ti-Sapphire laser oscillator as well as put in more GVD compensation devices such as prism compressors and chirped mirrors. The ﬁnal goal is to push to the Fourier limit, which gives us 59.6 fs pulse duration for 17 nm bandwidth. The energy scale of our analyzer has not been fully calibrated. We will use the Fermi edge in angle-integrated photoemission to calibrate this energy scale. Last but not the least, our chamber pressure is not ideal. It was only low 10−10-Torr-scale when we conducted the Sb testing experiment. We will have to conduct a careful bake-out of the whole chamber in order to achieve a low 10−11-Torr pressure.

7 Conclusion

Time-Resolved ARPES system is a powerful tool in studying ultrafast electron dynamics in correlated materials. In this thesis, I give a brief review of theoretical and experimental backgrounds for building such a system. I report the construction of a Femtosecond Time- Resolved ARPES system comprised of an ultrafast pump-probe laser output and an electron analyzer as its key components. An optical autocorrelation system verifies the time resolution to be 75 fs; a testing ARPES experiment on Sb(111) verifies the angular and energy resolutions to be less than 0.34◦ and 50 meV, respectively. The system is going through final optimization and maintenance process. The upcoming data production will disclose a complete image of ultrafast electron dynamics in correlated materials.

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