Excitons in Cuprous Oxide Fishman, Dmitry

University of Groningen

Excitons in cuprous oxide Fishman, Dmitry

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© Dmitry A. Fishman, 2008. Zernike Institute for Advanced Materials Ph.D.-thesis series 2008-10 ISSN 1570-1530 ISBN 978-90-367-3435-6 The work presented in this thesis was performed at the Optical Condensed Matter Physics Laboratory (University of Groningen). The project was supported by Zernike Institute for Advanced Materials, University of Groningen.

Front cover illustration by Murad A. Ibatullin ©

RIJKSUNIVERSITEIT GRONINGEN

Excitons in cuprous oxide

Proefschrift

ter verkrijging van het doctoraat in de Wiskunde en Natuurwetenschappen aan de Rijksuniversiteit Groningen op gezag van de Rector Magnificus, dr. F. Zwarts, in het openbaar te verdedigen op maandag 9 juni 2008 om 13:15 uur

door

Dmitry Fishman

geboren op 1 juli 1981 te Kazan, Sovjet-Unie

2 Promotor: Prof. dr. ir. P.H.M. van Loosdrecht

Beoordelingscommissie : Prof. dr. M. Salakhov Prof. dr. M. Gonokami Prof. dr. M. Fiebig

…to my parents

4 Content

Introduction 7

Chapter 1. Excitons in Cuprous oxide 10 1.1. General description of excitons 11 1.2. Exciton gas as a Bose gas 14 1.3. Excitons in cuprous oxide 17

Chapter 2. Paraexcitons versus orthoexcitons 26 2.1. Orthoexciton gas 27 2.1.1. Time evolution of the orthoexciton gas 27 2.1.2. Mechanisms of orthoexciton loss 31 2.2. Paraexciton lifetime: observation of unobservable 36 2.2.1. Intraexcitonic transitions 37 2.2.1.1. Principles of the ultrafast pump-probe experiment 39 2.2.1.2. Experimental details 42 2.2.2. Thermally-induced up-conversion 46 2.2.1.2. Principle of the experiment 46 2.2.2.2. Experimental details 47 2.2.3. Lifetime and discussion of exciton loss processes 49 2.2.4. Mechanisms of paraexciton losses 53 2.2.5. Exciton trapping by the crystal imperfections 59 2.2.6. Time evolution of paraexciton gas statistical parameters 70 2.2.7. Paraexciton gas parameters evolution in ( µTn )-space 78 2.2. Conclusions 79

Chapter 3. Exciton gas in a high magnetic field 84 3.1. Magneto-luminescence of exciton gas 85 3.1.1. Experimental details 85 3.1.2. Experimental results 86 3.1.3. Paraexciton kinetic energy distribution 91 3.1.4. Time-resolved magneto-luminescence experiments 92 3.2. Magneto-absorption of the exciton 95 3.2.1. Experimental details 95 3.2.2. Results and discussion 96 3.3. Conclusion 107

Chapter 4. Induced terahertz response in Cu 2O 110 4.1. Terahertz time-domain spectroscopy 111

5 4.1.1. Generation and detection of picosecond terahertz pulse 112 4.1.2. Experimental setup for terahertz time-domain spectroscopy 115 4.1.3. Propagation of an electromagnetic wave packet through the medium. 118 Terahertz spectral analysis 4.1.4. Time-resolved optical pump terahertz probe spectroscopy 125 4.2. Induced terahertz response in Cu 2O 129 4.2.1. Power dependence 134 4.3. Preliminary discussion 138 4.3.1. Possible theoretical considerations 138 4.3.2. Mathematical description: Drude+Lorentz model 141 4.4. Conclusion 145

Conclusions 148

Samenvatting 150

Acknowledgements 153

6 Introduction

In 1931 Yakov I. Frenkel proposed that it would be possible to excite charges in materials without influencing their electrical conductivity [1]. This suggestion can be marked as the birth of the exciton [1, 2], which turned out to be crucial in understanding many of the optical properties of condensed matter. The exciton can imagined as the electron-hole analog of a hydrogen atom, where the light hole takes over the role of massive positron. More than 20 years later, there were two independent experimental observations of a hydrogen-like absorption spectrum in Cu 2O in visible energy range by Gross et al. [3], and Hayasi et al. [4]. This was the first experimental proof of the actual existence of excitons, even though at the time Hayasi et al. did not relate the observed absorption lines to the exciton suggested by Frenkel. Now, we are many years later, and a vast experimental literature of precise experiments on excitonic properties of many materials exists. Arguably, the best documented case is Cu 2O. All classes of transitions predicted by the theory for exciton spectra are observed in different parts of the, mostly visible, energy region have been observed in this compound. It might therefore not be surprising that cuprous oxide was the favorite semiconductor among many physicists in the pre-silicon era. There are a number of well established phenomena which evidence that composite bosons made of even number of fermions, such as for example He-4 atoms, Cooper pairs, or alkali-metal atoms, behave under some conditions like ideal bosons showing a spectacular phase of large ensemble of these later particles: a macroscopic quantum state known as Bose-Einstein condensate. The exciton, being composed of two fermions, is an integral-spin particle, i.e. a composite boson. As a many particle system, these composite bosons show a variety of intriguing high density phenomena, including formation of excitonic molecules [5], electron-hole droplet and plasma formation [6], and even Bose- Einstein condensation [7]. Strong indications exist that Bose-Einsten condensation of excitons, predicted a long time ago [8], indeed occurs but, weirdly and wonderfully enough, in a peculiar system of the bilayer two-dimensional gases [9] in which excitons are formed under the equilibrium condition. The appearance of Bose-Einstein condensation of optically-created, finite-lifetime excitons is less evident [10] as only complex systems such as, for example, excitonic polaritons in semiconductor microcavities [11] provide signatures of a condensate phase. Since excitons in Cu 2O are strongly bound their bosonic character persists up to high densities and, due light masses of particles, their condensate form is expected at appealingly high temperatures [7]. These are the lowest energy state excitons (singlet or so-called paraexcitons of the yellow series) that are expected to condense. Remarkable, these excitons are optically inactive and therefore characterized of

7 very long lifetimes provided the crystal is of a perfect quality. Their significant population can be achieved via laser pumping of higher-energy, optically-active states and the subsequent efficient relaxation processes towards the lowest singlet state. Advantageous on one side, the optical darkness of singlet excitons does not, however, allow to easily probing their properties and this has been one of the main obstacles in searching of their possible condensate form.

This work revisits the excitonic properties of Cu 2O as observed with (time resolved) optical techniques, making use of current days experimental capabilities providing very precise measurements, with high energy and/or time resolution, using a variety of high power coherent pulsed light sources, intense magnetic fields, and cryogenic temperatures.. Besides short reviews of what has done in the past, a number of new experiments are discussed aiming at the determination of the exciton lifetime and the time evolution of statistical properties (temperature, density, chemical potential) of an exciton gas created by a short light pulse. In doing so, the reader is also introduced to the principles of ultrafast time-resolved spectroscopy and time-resolved luminescence spectroscopy. Particular attention will be paid to the question whether it is possible to create the proper conditions for an excitonic Bose-Einstein condensate. The presence of an external magnetic field drastically changes the properties of the excitons, leading for instance to fairly complicated excitonic magneto-absorption spectra, which are discussed in terms of a model which is also applicable to hydrogen in the presence of extremely high magnetic fields as found in the atmosphere of neutron stars [12]. One other consequence of the presence of a magnetic field is that excitons which are not active in optical experiments may be activated by field induced mixing with optically active excitons. Since the optically inactive excitons generally have a very long recombination time, ranging up to microseconds, this is of particular interest in view of a possible Bose-Einstein condensation of excitons. Finally, the last chapter discusses some surprising observations in the optical properties of Cu 2O in visible-pump, far infra-red (about 1 THz) probe experiments are discussed, which deviate substantially from the usual Drude like behaviour observed in semiconductors.

8 References

1. J. Frenkel, Phys. Rev. , 37 , 1276 (1931) 2. J. Frenkel, Soviet Experimental and Theoretical Physics , 6, 647 (1936) 3. E.F. Gross and N.A. Karryev, Doklady Akademii Nauk SSSR , 84 , 261 (1952) 4. M. Hayashi, J. Fac. Sci. Hokkaido Univ. , 4, 107 (1952) 5. N. Nagai, R. Shimano, and M.K. Gonokami, Physical Review Letters , 86 , 5795 (2001) 6. M. Nagai and M. Kuwata-Gonokami, J. of Luminescence , 100 , 233-242 (2002) 7. S.A. Moskalenko and D.W. Snoke, Bose-Einstein Condensation of excitons and biexcitons . 2000: Cambridge University Press. 8. L.V. Keldysh and A.N. Kozlov, Sov. Physics Solid State , 6, 2219 (1965) 9. J.P. Eisenshtein and A.H. MacDonald, Nature , 432 , 691-694 (2004) 10. D.W.Snoke and J.P.Wolfe, Physical Review B , 42 , (1990) 11. J. Kasprzak, M. Richard, S. Kundermann, A. Baas, P. Jeambrun, J.M. Keeling, F.M. Marchetti, M.H. Szymanska, R. Andre, J.L. Staehli, V. Savona, P.B. Littlewood, B. Deveaud, and le S. Dang, Nature , 443 , (2006) 12. D. Lai, E. E. Salpeter, and S. L. Shapiro, Phys. Rev. A , 45 , 4832-4847 (1992)

9 Chapter 1

Excitons in cuprous oxide

This chapter is devoted to a general description of excitons. One of the keynotes of the chapter is excitonic Bose-Einstein condensation: when the gas, consisting of bosons is at sufficiently low temperature all the particles condense into the lowest accessible quantum state forming a single macroscopic quantum state, resulting in a new form of a matter.

In addition, the optical properties of excitons in cuprous oxide (Cu 2O) created by photoexcitation are discussed. The absorption spectra show that the excitons in Cu 2O can be well understood in terms of the hydrogen atom model with only a slight modification. Finally, it is shown how one can retrieve information on thermodynamic gas parameters from the luminescence spectra.

10 1.1. General description of excitons.

As mentioned in the introduction, the possibility of the excitation of charges in a crystal lattice, which is not connected with the electrical conductivity of the crystal, was proposed by Yakov I. Frenkel in 1931, [1]. Frenkel suggested that photon absorption may lead to a charge neutral quasi-particle excitation of the electronic system: the exciton, [1, 2]. Using an incident photon, one can excite an electron in a semiconductor to the conduction band, leaving a hole in the valence band. This electron-hole pair may be bounded by Coulomb interaction to form an integer spin particle, called exciton. This system can be modeled as a hydrogen-like atom in which the relative motion of the electron and the hole can be described by an excitonic Bohr radius which is proportional to the square of the principle quantum number. For an excitonic system with elementary charges e in a medium with dielectric constant ε, and reduced mass µ, the ground state Bohr radius is given by:

h2 aB = 2 , (1.1) µ e ()ε

There are two types of excitons which are characterized by the ratio between the excitonic Bohr radius aB and the lattice constant l: Frenkel and Wannier-Mott excitons. Frenkel excitons have a Bohr radius of the order of the lattice constant or smaller. Such exciton is strongly bound and usually localized on one site. The electron and the hole do not move independently. In contrast, a the Bohr radius of a Wannier-Mott exciton is much larger than the lattice constant, therefore it has a smaller binding energy and is delocalized over a number of sites. Both the electron and the hole are mobile. For this type of exciton there are atoms inside the exciton orbit which cause the screening of the Coulomb interaction to –e2/εr2, leading, for instance, Eq. 1.1. In the excitonic system, the Coulomb interaction binding the electron and the hole decreases the energy of the pair compare to the non-interacting pair. The electron and the * * hole can be treated as two interacting particles with masses me , mh , respectively. The Hamiltonian can be written as [3]:

2 2 2 2 2 2 h ∇e h ∇h e h l(l +1) H = − * − * - + 2 , (1.2) 2me 2mh ε re − rh 2µ r

11 where ε is the lattice dielectric constant, l is the angular momentum quantum number and re and rh are the positions of the electron and the hole, respectively. The corresponding eigen values form a series of exciton energies given by:

2 2 e 2 2 2 µ( ) h k h l(l +1) E = E − ε + + , (1.3) n G 2h2n2 2M 2µ r 2

1 1 1 * * = * + * M = me + mh µ me mh

Here EG is the band-gap energy, the second term is the exciton binding energy and the third term is the kinetic energy of exciton of a wave vector k. From the Eq. 1.3, a series of sharp peaks can be expected in the exciton absorption spectra (that is, when optical transitions are allowed, see §1.2). The exciton creation can be described in terms of a band theory. A schematic diagram of an exciton creation and recombination processes is shown in Fig. 1.1. A pure semiconductor has, at zero temperature, no mobile charge carriers. The valence band states are completely filled with electrons and the conduction band is completely empty. Photons with energy hν exeeding the semiconductor band-gap EG, create non-equilibrium electrons in the conduction band and the holes in the valence band which can bind into excitons (channel A on Fig. 1.1). Excitons can also be formed directly via the transitions from the valence band to the exciton states (channel B on Fig. 1.1). Once formed in a relatively pure crystal, the excitons behave as mobile particles with a finite lifetime. The thermal relaxation as well as the diffusivity of excitons will be discussed in §2.2. Excitons may make themselves visible by decaying. The radiative recombination of the electron and the hole back to the ground-state (channel C on Fig. 1.1) produces a photon with an energy:

hν = EG − Eex , (1.4) Moreover, the recombination may also take place together with the phonon assistance (channel D):

hν = EG − Eex + Ek ± Ep , (1.5)

where Ek is the exciton kinetic energy and E p is the energy of the phonon and “+” and “-“ corresponds to anti-Stokes and Stokes processes, respectively. Momentum conservation requires:

Fig. 1.1. Different ways of exciton creation are possible: process A, an electron is excited from the valence band to a conduction band. It may decay either radiatively or non- radiatively to an excitonic state. Process B, direct exciton creation. Process C, direct exciton recombinatio. Process D, phonon-assisted recombination.

r r k = k photon (direct tr ansition) r r r k = k photon + k phonon (phonon - assisted), (1.6)

The direct process can only occur for an exciton with a momentum matching that of the emitted photon, yielding a sharp luminescence line at nearly zero kinetic energy. The phonon-assisted process, on the other hand, samples all of the occupied exciton states for the exciton momentum is easily absorbed by the optical phonon. It is important to note that the symmetries of the electronic and the vibrational states are crucial to the relaxation properties of the exciton – in particular, their lifetime and scattering rates. A detailed discussion of this can be found in §3.1.3, in particular for the exciton transitions in cuprous oxide.

13 3.2. Exciton gas as a Bose gas.

The exciton, being composed of two fermions, is an integral-spin particle, i.e. a composite boson. The quantum statistical rules for a boson is that its probability of being scattered into the state of wavevector k is proportional to 1+f k, where fk is the occupation number of the state. In other words, the likelyhood of a boson being scattered into a given state is enhanced by the other bosons already occupying that state. It can be easily shown that this “stimulated” scattering leads to the Bose-Einstein distribution function:

1 r fk = f (E) = E −µ , (1.7) e k BT −1 where E= ħ2k2/2m is the kinetic energy of an exciton and µ is the chemical potential, which is measured with respect to the total kinetic energy of the exciton at E=0 and determined by the condition:

f r = N, (1.8) ∑r k k where N is the total number of excitons in the system. For a macroscopic volume, the number of plane-wave states per unit energy for the ideal non-interacting gas has the form D(E)=CE 1/2 . Therefore, the number of excitons per unit energy is given by the product of the density of states and the occupation number f(E) :

1 E 2 N(E) = C E −µ , (1.9) e k BT −1

If N excitons occupy a volume V, then we can determine C from the usual counting of k states, [4]:

3 gV  2m  2 C =   , (1.10) 4π 2  h2  where g is the spin multiplicity. The constraint Eq. 1.8 may be re-stated as:

N = ∫ N(E)dE , (1.11)

In terms of the gas density, n=N/V, we can, finally, write:

3 1 g  2m  2 ∞ E 2 n = 2  2  ∫ E −µ dE , (1.12) 4π  h  k T 0 e B −1

For a given density and temperature, Eq. 1.12 fixes the chemical potential m of the exciton gas. By defining ε=E/k BT and the dimensionless chemical potential α=-µ/ k BT, Eq. 1.12 can be written in terms of a dimensionless integral as:

1 2 ∞ ε 2 n = nQ 1 ∫ ε +α dε 2 π 0 e −1 3  mk T  2 n = g B  , (1.13) Q  2πh2 

where nQ is the “quantum density”, at which the thermal de’Broglie wavelength of the particles equals the inter-particle spacing. Several examples of the shape of the kinetic energy distribution for a given µ and T are presented in Fig. 1.2. In Fig. 1.2 we plot Eq. 1.9 at a fixed temperature T as the density of the gas is hypothetically increased. At low densities, the chemical potential is large and negative. This reduces the distribution in Eq. 1.9 to the density-independent shape given classical Maxwell-Boltzman distribution. As we increase n, the chemical potential approaches zero: the spectrum becomes more sharply peaked. Surprising things happen when more particles are added to the system at a fixed temperature. At some point the density will be reached where µ becomes 0. The integral in Eq. 1.12 is still finite. In ideal system, the number of particles in the ground state is proportional to –g/ α, where α is a dimensionless chemical potential. For a realistic volume, the first excited k-state will also have a large occupation number, but considerably fewer particles than the ground state, due to the sharply peaked distribution function. This effect in which thermodynamic equilibrium of the ideal Bose gas forces a macroscopic number of particles into the lowest state is the Bose-Einstein condensation (BEC).

Fig. 1.2. The shape of the kinetic energy distribution at a fixed temperature (T=10K) but for a different chemical potential values.

The phase boundary for BEC of the Bose gas is determined by setting chemical potential in Eq. 1.12 to zero, yielding:

3  mk  2 n = ,2 612 n = ,2 612 g B T  , (1.14) BEC Q  2πh2 

It is obvious from Eq. 1.14, that for a given gas density, Bose-Einstein condensation is expected to be reached at much higher temperatures in an exciton system than in a gas of atoms, due to the light mass of the excitons. Theories of excitonic condensation (and superfluidity) have been given by Keldysh and Kozlov [5], Haug and Hanamura [6]. The theory of Keldysh and Kozlov [5] discusses the system of the degenerate electrons and the holes of the arbitrary density. At high densities the gas-liquid type phase transition of exciton gas into an electron-hole Fermi liquid is possible - the bound state of a macroscopically large number of electrons and holes, [7-10]. The electron-hole liquid is similar to the metallic hydrogen or alkali metals. Unlike the

16 common metals, in the electron-hole liquid not only the electrons but also the holes are Fermi degenerate. At low temperatures the electron-hole liquid transforms to an “excitonic insulator” phase, [5, 11, 12]. In this two-Fermi-liquid state the collective pairing of the electrons and the holes in the vicinity of Fermi surfaces arises. This process is similar to the Bardeen-Cooper-Schreiffer pairing in superconductors (BCS state). This pairing should manifest itself in the appearance of the energy gap around the Fermi surface. In that sense, the excitonic insulator state in the non-equilibrium electron-hole system is a coherent BEC state of the high-density excitons, as the superconducting state is a coherent BEC state of Cooper pairs. Haug and Hanamura [6] treated the “condensation problem” in the spirit of the Bogoliubov’s model in which excitons are viewed as approximate point bosons with a some interaction with each other. Several assumptions are implicit in this model. Work of Haug and Hamura based on assumption that the excitons have repulsive interaction, otherwise, at increasing densities excitons aggregate into a complex system. As an example, the same problem prevents BEC of a normal gas of hydrogen molecules. Further, the lifetime of the excitons should be longer than time required to establish a true equilibrium with a lattice. And finally, the last assumption is in a negligible interaction between excitons and the radiation field. If the exciton-photon coupling is strong, then the true elementary electronic excitations of the crystals are better described in terms of polaritons [13].

3.3. Exciton gas in cuprous oxide.

The optical spectrum of Cu 2O is remarkable since all classes of transitions predicted by the theory for exciton spectra are observed in different parts of the energy region. It might therefore not be surprising that cuprous oxide was the favorite semiconductor among many physicists in the pre-silicon era. Cuprous oxide crystallizes in a cubic lattice (lattice 4 constant of 4,3Å) with two molecules per unit cell and space group symmetry O h . The side symmetries of copper and oxygen are D 3d and T d, respectively, [14]. For convenience the crystal structure is presented in Fig. 1.3a. The band structure of Cu 2O contains in total 10 valence bands and 4 conduction bands (Fig. 1.3b shows the lowest two conduction bands, and the highest two valence bands at the zone center), [15]. The direct energy gap between the highest valence and lowest conduction bands is 2,17eV (at T=10K), [16, 17]. The + + lowest conduction band Γ6 is formed by Cu 4s orbitals and the highest valence band Γ7 is formed by Cu 3d orbitals. Holes and the electrons in these four bands interact through a screened Coulomb interaction and form exciton series in the energy range of visible light.

Fig. 1.3. (a) Crystal structure of Cu 2O. Dark grey balls represent oxygen and light grey one

- copper. (b) Energy band structure of Cu 2O near the zone center. The exciton 1s level lays 150 meV below the first conduction band.

The yellow series are derived from the highest valence band and the lowest conduction band ( V1-C1). The green series is related to the gap V2-C1, the blue series to V1-C2 and the indigo series to the gap V2-C2. For the yellow series the energy gap is direct, but dipole forbidden. To see if a transition can be made we should calculate the absorption coefficient:

π π α ∝ ∑ Γf Di Γi , (1.15) i where Γf and ΓI correspond to the final and the initial states, respectively, with parity π, and 2 Di proportional to pi for the dipole operator and to pi for the quadrupole operator, [18]. For the Cu 2O the lowest conduction band and the highest valence band have the same positive parity. The dipole operator has a negative parity. When we integrate over the space, the absorption coefficient will therefore always be zero. The quadrupole operator has a positive parity. This means that the matrix element can be non-zero, which is the case for the V1-C1 transition. For excitonic levels the total parity is determined by the product of the parities of the valence band, the conduction band and the exciton envelope wave function. Since both

18 valence and conduction bands have a positive parity, the total parity is determined by the parity of the exciton level. For an excitonic s-state, the parity is positive, and the transition to the ground state is dipole forbidden, but quadrupole allowed. For a p-state, the parity is negative, and the transition to the ground state is dipole allowed. Due to spin-orbit interaction, every excitonic level is split into a triplet ortho level (J=1) and a singlet para level (J=0). The corresponding wave functions of the ortho and para levels can be written as follows: ortho para

O−1 = ↓e ↓h

O0 = ()↑e ↓h + ↓e ↑h / 2 P = ()↑e ↓h − ↓e ↑h / ,2 (1.16)

O1 = ↑e ↑h

For the 1s exciton state the ortho-paraexciton energy splitting is 12 meV. Because only transitions between levels can be made when ∆J=±1, decay from any paraexciton state to the ground state is forbidden for all orders of perturbation. This suggests that theoretically paraexcitons can live very long. Practically, the lifetime is limited by the defects and impurities (§2.2 and references therein). For 1s orthoexcitons decay to the ground state is quadrupole allowed. However, the lifetime of 1s orthoexcitons is mainly limited by transition to the lower lying paraexciton state and reveal a value around 1,2 ns (§2.1 and references therein). Almost 60 years ago Hayashi [19, 20] observed a series of hydrogen-like lines in the absorption spectrum of solid cuprous oxide which was cooled below the room temperature. During the same time investigators from Leningrad (St-Petersburg) [21-23] and Strasbourg [24-27] made more accurate measurements of the wavelengths of these lines over a wide range of temperatures.

The absorption spectrum of the Cu 2O oxide crystal (in [100] crystallographic direction) at T=1,2K is shown in Fig.1.4. Several absorption lines corresponding to transitions to the ortho exciton p-states are clearly observed. All lines are superimposed on a background due to the tail of the phonon assisted absorption to the conduction band continuum. The spectral positions of the observed absorption lines are in good agreement the Rydberg equation (Eq. 1.17, see inset Fig. 1.4). The estimated Rydberg constant for the exciton with reduced mass

µ=0,34 me [22, 23] in a medium with dielectric constant ε=7,2, [28] Ry=94 meV is close to the one, obtained from the experiment Ry=97,6 meV.

Fig. 2.4. Optical absorption versus photon energy in cuprous oxide at 1,2K, showing a

series of exciton lines for Cu 2O sample of [100] crystal orientation (sample was provided by A. Revcolevschi, University of Paris IV). Inset: empirical fit to the absorption peak position.

2 2 µ(e ) E = E − ε , (1.17) n G 2h2n2

where EG is the V 1-C1 band gap at T=1,2 K. As mentioned above, the n=1 level can not be observed in absorption spectrum, since the transition to this state is only quadrupole allowed. However, one can easily observe n=1 orthoexciton luminescence (Fig. 1.5). The n=1 level energy position deviates from the hydrogen model. The n=1 exciton Bohr radius (7Å) is comparable with the lattice constant (4Å): the screening of the interaction is different then for the higher states giving the binding energy of 150 meV (binding energy at n=2 state is 90 meV).

20 The luminescence spectrum of Cu 2O at T=1,2 K is presented in Fig. 3.5. The sharp band (2,035 eV) is due to direct recombination of the orthoexciton. The broad spectral lines are phonon-assisted orthoexciton lines. Since there are several optical phonons with the

Fig. 1.5. Luminescence spectra of excitons in Cu 2O for [100] orientation lines for Cu 2O sample of [100] crystal orientation (sample was provided by A. Revcolevschi, University of Paris IV) at T=1,2K. appropriate symmetry to participate in the recombination process, each having a different energy, there are several phonon replicas observed in the luminescence spectrum. There are + clear lines from phonon-assisted orthoexciton ( Γ5 ) transitions with the simultaneous - - - excitation of Γ5 (2,024 eV), Γ3 (2,026), and Γ4 (2,020) phonons, [29]. As it mentioned in §1.1, the phonon-assisted process samples all of the occupied exciton states, because the exciton momentum is easily absorbed by the optical phonon. The optical phonon is slightly dependent on the phonon wave vector. For the typical gas temperature of 100 K or less, the exciton wave vectors are small ( k<10 7 cm -1) compared to the Brillouin-zone boundary (~10 8 cm -1), [30]. The optical-phonon dispersion, as measured by neutron scattering [31], is negligible for these wave vectors. Therefore, the phonon-

21 assisted processes give an accurate replica of the energy distribution of the excitons. The observed emission band can be described by: where D(E)~E 1/2 is the density of exciton states (assuming a quadratic dispersion), and

∞ (x−E )2 − I(E′) = A ∫ D(x) f (x)e Γ 2 dx , (1.18) −∞ f(E) is either the Maxwell-Boltzmann (classical gas) or Bose-Einstein (quantum gas) distribution function. Γ represents the spectral resolution of the experimental setup. Fitting this equation to the experimentally observed spectrum yields an estimate for the effective exciton temperature and chemical potential. For an ideal, non-condensed, Bose gas the density of bosons is given by Eq. 1.12.

A precise determination of the Cu 2O samples chemical composition [32] shows that the amount of oxygen and copper usually deviates from a perfect stoichiometric composition. In other words, the samples will contain a certain number of copper and oxygen vacancies. The presence of these vacancies results in three characteristic luminescence bands (Fig. 1.6a): one band centered around 1,35 eV which is attributed to copper vacancies, the two higher energy bands centered around 1,71 eV and 1,51 eV are attributed to oxygen vacancies [32, 33].

Fig. 1.6. (a) Vacancy spectrum of Cu 2O at T=77K (b) Electronic energy structure of Cu 2O of [100] orientaton including vacancy states (sample was provided by A. Revcolevschi, University of Paris IV). The excitation wavelength 532 nm (2,33 eV).

22 Oxygen vacancies V O can exist in Cu 2O in three states: V O2+ , an unoccupied vacancy with a double positive charge with respect to the lattice; V O2+ +e, a one-electron vacancy, positively charged with respect to the lattice; and V O2+ +e+e, a two-electron vacancy, neutral with respect to the lattice. The ground levels of these vacancies are situated above the valence band, in the forbidden energy gap, in the order indicated in Fig. 1.6b. It is assumed that the higher energy luminescence band is originating from the V O2+ +e state. Within the described picture, the possibility of conversion from one type of center into another as a result of two simple processes: the trapping of either electrons or holes from the valence band. The luminescence line at 1,35 eV (Fig. 1.6a) originates from transitions from the copper vacancy V Cu’ which is negatively charged with respect to the lattice. The purity of the sample (amount of the vacancy centers) is very critical for certain properties of the exciton gas, in particular to the lifetime of the excitons, as is discussed in §2.2.

23 References

1. J. Frenkel, Phys. Rev. , 37 , 1276 (1931) 2. J. Frenkel, Soviet Experimental and Theoretical Physics , 6, 647 (1936) 3. K. Karpinska, P.H.M. van Loosdrecht, I.P. Handayani, and A. Revcolevschi, J. of Luminescence , 112, 17-20 (2005) 4. F.K. Richtmyer, E.H. Kennard, and J.N. Cooper, Introduction to Modern Physics . 6 ed. 1969, New York: McGraw-Hill. 5. L.V. Keldysh and A.N. Kozlov, Sov. Physics Solid State , 6, 2219 (1965) 6. E. Hanamura and H. Haug, Physics Reports C , 33 , 209 (1979) 7. L.V. Keldysh, Electron-hole liquid in semiconductors . Modern Problems of Condensed Matter Science, ed. C.D. Jeffries and L.V. Keldysh. Vol. 6. 1987: North-Holland, Amsterdam. 8. L.V. Keldysh, Contemp. Phys. , 27 , 395 (1986) 9. T.M. Rice, Solid State Physics , 32 , 1 (1977) 10. L.V. Keldysh. in 9th Int. Conf. on Physics of Semiconductors . 1968. Moscow. 11. B.I. Halperin and T.M. Rice, Solid State Physics , 21 , 115 (1968) 12. J. des Cloizeaux, J. Phys. Chem Solids , 26 , 259 (1965) 13. J.J. Hopfield, Phys. Rev. , 112 , 1555 (1958) 14. C. van der Vegte, Exciting exciton . 2000. 15. J. Ghijsen, L.H. Tjeng, J. van Elp, J. Westerink, G.A. Sawatzky, and M.T. Czyzyk, Phys. Rev. B , 38 , 11322-11330 (1988) 16. A.Jolk, M.Jorger, and C.Klingshirn, J. Phys. Rev. B , 65 , 245209 (2002) 17. D. Frohlich and R. Kenklies, Phys. Stat. Sol. (b) , 111 , 247 (1982) 18. N. Caswell and P.Y. Yu, Phys. Rev. B , 25 , 5519-5522 (1982) 19. M Hayashi and K. Katsuki, J. Phys. Soc. Japan , 7, 599 (1952) 20. M. Hayashi, J. Fac. Sci. Hokkaido Univ. , 4, 107 (1952) 21. E.F. Gross, Izvest. Akad. Nauk SSSR Ser. Fiz. , 20 , 89 (1956) 22. E. Gross and B.P. Zakharchenia, Doklady Akad. Nauk SSSR , 90 , 745 (1953) 23. E. Gross, B.P. Zakharchenia, and N.M. Reinov, Doklady Akad. Nauk SSSR , 92 , 265 (1953) 24. S. Nikitine, Phil. Mag. , 4, 1 (1959) 25. S. Nikitine, L. Couture, G. Perny, and M. Sieskind, J. Phys. Radium , 16 , 425 (1955) 26. S. Nikitine, Helv. Phys. Acta , 28 , 307 (1955) 27. S. Nikitine, G. Perny, and M. Seiskind, Compt. rend. , 238 , 67 (1954) 28. M. Jorger, T. Fleck, C. Klingshirn, and R. von Baltz, Physical Review B , 71 , 235210 (2005)

24 29. Ed. J.T. Devreese, Polarons in Ionic Crystals and Polar Semiconductors. 1972. 30. S.A. Moskalenko and D.W. Snoke, Bose-Einstein Condensation of excitons and biexcitons . 2000: Cambridge University Press. 31. M.M. Beg and S.M. Shapiro, Phys. Rev. B , 13 , 1728 (1976) 32. J. Bloem, Phil. Res. Rep. , 13 , 2 (1958) 33. N.A. Tolstoi and V.A. Bonch-Bruevich, Soviet Physics Solid State , 13 , 1135-1137 (1971)

25 Chapter 2

Paraexcitons versus Orthoexcitons in a quest for Bose-Einstein condensation

In order to see whether one can make an excitonic Bose-Einstein condensate, one firstly needs to have a look at the statistical gas parameters. As an example, the lifetime is considered to be an extremely important parameter. It should be long enough to keep a sufficient density while at the same time thermalization with the lattice should occur fast enough to reach the critical quantum density. Since both types of excitons discussed in Chapter 1 reveal bosonic behavior, both of them can theoretically form a condensate. In this chapter we discuss which one is the best candidate. In this chapter we present a few experiments aiming at the determination of 1s orthoexciton and paraexciton gas parameters. We focus mainly on the determination of important parameters as the gas temperature, the chemical potential and the population dynamics. In addition, the reader will be introduced to the principles of ultrafast time- resolved spectroscopy and time-resolved luminescence spectroscopy.

26 2.1. Orthoexciton gas.

Since the orthoexciton transition from the 1s excited state to the ground state is quadrupole allowed, it is relatively easy to study this gas using luminescence spectroscopic techniques. Due to this, orthoexcitons have been studied intensively over the past few decades [1, 2] . In this section we will discuss the results of the one of the recent optical studies, more fully described in K. Karpinska et al. , [3]. The main goal of this investigation was to establish the statistical gas parameters and to obtain some information on the temporal evolution of the orthoexciton gas after initial excitation using time-resolved experiments. Some of the experimental techniques are discussed later on in section 2.2. Here we focus mainly on the results with only brief references to the key experimental parameters.

2.1.1. Time evolution of the orthoexciton gas.

Karpinska et al. [3] performed time-resolved experiments using a traveling wave optical parametric amplifier pumped by 1kHz, 1.55 eV, 120 fs pulses produced by an amplified Ti:Sapphire laser system to excite the excitons. The 150 fs excitation pulses (1.4 – 2.5 eV) were focused on a sample with a spot size of about 100 µm. The exciton emission was detected by a Hamamatsu streak camera system operating in photon counting mode with a temporal resolution of 10 ps. The experimental spectral resolution function is measured to be of a Gaussian type with a full width at half height Γ = 0.37 nm. Figure 2.1 shows the time evolution orthoexciton luminescence spectrum after excitation with a 150 fs pulse of 1.4 eV energy at T = 7 K. Initially, a broad spectrum is observed dominated by the phonon-assisted transitions reflecting the occupation of the excitons in the higher levels. Despite the fact that the excitation was to a higher lying band, [3], which is a rather indirect way of excitation, no initial growth of the luminescence has been observed, and the exciton creation appears to be instantaneous upon excitation within the experimental resolution of 10 ps. As time progresses both the shape and the integrated intensity of the emission change. The former reflects the cooling of the exciton gas and thus the relaxation of the distribution of occupied states, whereas the latter reflects the temporal decay of the total exciton population. The direct and phonon-assisted peaks become well resolved for delay times longer than 200 ps, and show similar long time decay dynamics.

Fig. 2.1. Time evolution of the orthoexciton emission spectrum after two-photon excitation with 1.4 eV photons. Cu2O crystal of [100] orientation. The lattice temperature is T = 7 K, [3].

As described in Chapter 3, one can determine the statistical gas parameters by fitting the phonon-assisted emission line with Eq. 1.18. The parameters resulting from the fitting of this function to the time-resolved spectra are shown in Fig. 2.2. The cooling of the gas, shown in Fig. 2.2a, shows a bi-exponential behavior with decay times τ ~ 0.2 ns and τ ~ 6 ns. The initial, fast, cooling occurs via optical phonon emission and down-conversion to the paraexciton state, [3]. Once the kinetic energy of the excitons becomes to low for this process, the relaxation occurs via acoustical phonon emission with a relatively long decay time. The chemical potential, shown in Fig. 2.2b, decays faster than the temperature, with a time constant τ ~ 1 ns. This decay is essentially due to the loss of particles. Consistently, the integrated intensity (Fig. 2.3) of the luminescence shows a similar decay time ( τ ~ 1.4 ns).

Fig. 2.2. The time dependence of the exciton gas temperature (a) and the chemical potential (b) as obtained from an analysis of the phonon-assisted emission line, depicted in Fig. 2.1, using Eq. 1.18 [3].

Fig. 2.3. Temporal decay of the integrated intensities of the total, direct (DI), and phonon- assisted (PA) luminescence. The solid line is a fit to a single exponential decay. Excitation energy 1.4 eV, T = 7 K [3].

29 The (n,T) phase diagram is shown in Fig. 2.4. The line represents the phase boundary, whereas the symbols represent the experimental data from Fig. 4.2 a,b (using Eq. 1.18 to determine the density). Unfortunately, the state of the gas follows the Bose-Einstein condensation phase boundary without ever crossing it. A similar result was observed in [4].

Fig. 2.4. (n, T) phase diagram. The solid line is the Bose-Einstein condensation boundary. n is calculated using Eq (1.18).

Different explanations have been proposed for this so-called quantum saturation effect. In [5] it has been suggested that the quantum saturation phenomenon originates from the fact that the orthoexciton gas is not in an equilibrium state so that the number of particles is not well defined. The results discussed here do not support this conclusion. The kinetic energy distribution of the orthoexciton gas can be described with a very good accuracy by the Bose distribution, and, moreover, the decay of the chemical potential coincides with the particle decay obtained independently from the experiment. Therefore, one may conclude that the orthoexciton gas is in fact in quasi-equilibrium, and the particle density is indeed a well defined number. A few other explanations have also been suggested: dominant Auger recombination and/or strong spatial inhomogeneity, [6]. Both these effects are expected have only a marginal influence on the decay kinetics. Finally, Ell et. al. , [7] presented an analysis of the orthoexciton relaxation including polaritonic effects. The main conclusion of that analysis is that the so-called polaritonic ‘bottle-neck’ effect prevents formation of an

30 orthoexcitonic Bose-Einstein condensate at k=0. However, as pointed out in [8] the authors assumed in their model an infinite volume and it is not clear to what extend their conclusions would change when the finite volume relevant to experiments is taken into account. Independent of the origin, the experiments clearly show that the gas adjusts its quantum properties to the density and temperature and that it is in quasi-equilibrium. Apparently the particle decay is faster than the cooling rate, which inhibits a BEC transition. The eager reader may find more discussion, and a comparison to the paraexciton density evolution at the end of the current chapter.

2.1.2. Mechanisms of orthoexciton loss.

Several mechanisms of exciton loss were proposed in literature [3, 9, 10]. It is important where the exciton gas was investigated, in the bulk or near the surface. Since the cubic symmetry near the surface is broken, one may expect additional mechanisms of particle decay. The loss processes described in this paragraph consider bulk excitons only. Some of the described mechanisms work for both types of excitons. However, here we mainly focus on orthoexciton gas. Some additional mechanisms relevant to paraexcitons are described in paragraph 2.2. Figure 2.5 depicts several processes which may influence the orthoexciton and paraexciton population: transition to the ground state with the emission of photon (panel 1), phonon-assisted orthoexciton down-conversion (panel 2), Auger processes for ortho- and paraexcitons (panel 3) and spin-flip scattering ortho paraexciton conversion (panel 4). Since the optical transition (Fig. 2.5, panel 1) to the ground state is only quadruple allowed, the transition rate is quite small, which means that the time constant should be longer than the one observed from experiment, [11]. The dominant relaxation channel is the phonon-assisted ortho para conversion (Fig. 2.5, panel 2). The Auger mechanism (Fig. 2.5, panel 3), in which the scattering of two excitons causes one to recombine and the other to excite to a higher lying level, was initially proposed by Hulin et al . [12] and is also discussed in for instance [13]. For an exciton-gas density n, the rate of decay is A times n, where A is the Auger constant. Considering only these two mechanisms and assuming a spatially uniform gas density, the initial decay rate of orthoexcitons is given by:

Fig. 2.5. Schematic diagram of ortho and paraexciton state interaction – possible processes of exciton losses. 1 – optical transition to the ground stat; 2 – phonon- assisted down-conversion process; 3 – Auger process; 4 – spin-flip scattering ortho →paraexciton conversion;

32 -1 where τi is the exciton lifetime due to recombination at impurities and D=1.2 ns is the phonon-assisted down-conversion rate into paraexcitons, which has been measured at low gas density [14]. Moreover, one must take into account the re-generation of ortho and paraexcitons by the same process. In general, due to the different spin states of 1s excitons, there are actually three possible Auger constants, corresponding to various collisions between ortho and paraexcitons. We can write these ‘spin-dependent’ Auger constants as

Aoo , Aop , and App . For simplicity we will assume that these are all equal. The rate equations for the ortho ( no) and para ( np) densities are given by:

dn O nO 3 2 = − − 2Ann O + An − Dn O , (2.2) dt τ i 4 dn P nP 1 2 = − − 2Ann P + An + Dn O , (2.3) dt τ i 4

where the total exciton density is n=n O+n P. On the right-hand side of each equation the second term represents pure Auger decay, and the third term represents re-generation of excitons from ionized e–h pairs. The factors 3/4 and 1/4 originate from the ortho and para degeneracies. The predicted value of Auger constant was around 10-16 cm 3/ns [6, 15, 16]. However, in several recent spectroscopic experiments the Auger constant was determined as A=2,2*10 -21 cm 3/ns [17] and A=3,5*10 -22 cm 3/ns [18, 19]. To elucidate the influence of the presence of Auger processes at high density on the orthoexciton decay we have performed luminescence experiments using intense cw excitation at 532 nm. Fig. 2.6 shows the power dependence of the integrated phonon assisted luminescence as function of the excitation power [20]. The intensity shows a sub- linear growth upon increasing excitation power, and finally saturates for powers exceeding 30 mW. The behavior prior to saturation is well described by a square root of the excitation power dependence (solid line in Fig. 2.6). Since the particles produced by the Auger process reform into both orthoexcitons and paraexcitons one should expect such a nonlinear behavior (see also Eq. 2.1). The same result was observed in [1, 17]: the intensity increases by the square root of the orthoexciton population.

Fig. 2.6. The intensity of phonon-assisted luminescence as a function of the excitation power. Excitation energy 2.4 eV, T=7 K, [20].

Recently, Kavoulakis and Mysyrowicz [11] have suggested that orthoexcitons at high densities primarily decay by a spin–flip processes (Fig. 2.5, panel 5) rather than by Auger processes. In this case, two orthoexcitons scatter and exchange an electron or a hole, producing two paraexcitons that subsequently release their excess kinetic energy by phonon emission. The decay of orthoexcitons by this ‘spin–flip scattering’ process (plus phonon- assisted down-conversion) is given by:

dn O = −2Bn 2 − Dn , (2.4) dt O O where B is the spin–flip constant. Two main factors distinguish this process from the Auger mechanism. First, this is a ‘one-way process’, because two relaxed paraexcitons do not have sufficient energy to produce two orthoexcitons. Second, the total exciton number is conserved. Kavoulakis and Mysyrowicz predicted the ortho-para spin-flip conversion scattering rate being B = 1.5·10 -16 cm 3/ns [11]. The ortho and para rate equations for spin– flip scattering are given by:

dn n O = − O − 2Bn 2 − Dn , (2.5) dt τ O O

dn P nP 2 = − + 2Bn O + Dn O , (2.6) dt τ

where we have again included the impurity-induced nonradiative recombination time, τi, which in natural-grown crystals is typically about 300 ns [14, 21]. Adding the two equations indeed shows that this process conserves the total exciton density (n O+n P) if one neglects the slow recombination at impurities. The relevance of spin-flip processes has been discussed by Kubouchi et al. [22] using mid-IR absorption experiments: the paraexciton buildup rate increased with excitation power, suggesting a spin–flip constant B=2.8·10 -15 cm 3/ns. Finally, there is a possibility that two orthoexcitons can combine to form an excitonic molecule, so-called bi-exciton. The four-particle bi-exciton (two electrons and two holes) would have a much shorter Auger lifetime than excitons in the relatively dilute gas.

However, bi-excitons have not been spectroscopically observed in Cu 2O [23], [24]. In conclusion, we discussed several mechanisms of orthoexciton loss (Fig. 2.5). Mainly, the orthoexciton population decay is driven by down-conversion to the paraexcitons at early times. The spin-flip scattering of orthoexcitons may be also effective at early times (few ns). The orthoexciton decay due to the Auger process of orthoexcitons at high densities should go in hundreds of ns and is not relevant to explain the present data. In addition, paraexciton Auger process can explain the re-generation of orthoexcitons at late times. However, the experimentally determined constants for this process are orders of magnitude below than predicted by theory. The capture of two excitons into a bi-exciton is expected to be much more effective than exciton-exciton Auger decay, which involves recombination.

2.2. Paraexciton lifetime: observation of an unobservable

As already mentioned in Chapter 3, the transition from the 1s paraexciton excited state to the ground state is forbidden for all orders of perturbation. This circumstance makes paraexcitons highly interesting from the solid state physicist’s point of view: the lifetime should now be long enough for sufficient gas cooling. On the another hand, it is highly unpleasant from the optical spectroscopist’s point: the detection of particles tends to be more difficult, since optical transition to the ground state is forbidden. A method to make the 1s paraexcitons optically active is to break the symmetry. These kind of experiments are interesting, since the luminescence spectrum yields direct information on the statistical distribution of occupied states. There are several methods to break the cubic symmetry in cuprous oxide. The most ‘popular’ and well-studied method is the application of external pressure [25-27]. Such measurements have played an important role in the study of Cu 2O. Gross and Kaplyanskii’s observation in 1960 [28] of the polarization of the stress-split components of the 1s yellow state established its quadruple character and provided the experimental basis for Elliot’s 1961 band assignment, [29]. However, the paraexciton lifetime, determined from these experiments can not be named intrinsic in the sense that the system was substantially changed and, as a consequence, the lifetime was reduced by orders of magnitude. The symmetry can also be broken by the application of the electric fields. Some experiments, as well as electronic band structure calculations in the presence of electric field were presented in [30, 31]. Finally, also an applied magnetic field breaks the symmetry [32-34]. In this case, the optically forbidden paraexciton state |S> mixes with the quadruple allowed m J=0 orthoexciton state |T 0>. This leads to a weak, field tunable emission from the paraexciton 2 2 state, proportional to (ge − g h ) B , while the radiative lifetime of the particles becomes inverse proportional to the B 2. More details are given in Chapter 1. Below we discuss a few methods for the determination of the paraexciton lifetime in an unperturbed medium, using a novel experimental technique. After a short introduction to the experimental idea and details, we discuss how one can obtain statistical gas parameters from the measured optical response and introduce the early time dynamics of a paraexciton gas.

36 2.2.1. Intraexcitonic transitions

As suggested by Haken [35] and Nikitine [36], it should be possible to detect transitions between the ground state of the exciton with principle quantum number n=1 and higher lying levels in excitonic series. This type of absorption, which changes the angular momentum of the relative electron-hole wave motion from l=0 (s) state to l=1 (p) states (Fig. 2.7), is the equivalent to the absorption lines in atomic hydrogen known as the Lyman series. The line profile for the interexcitonic transitions has been recently calculated by Johnsen and Kavoulakis [37]. These authors were the first to point out that the absorption profile should provide direct information on the energy distribution of excitons occupying the n=1 state. In particular, in the presence of a Bose-Einstein condensation of excitons, the 1s-2p line should undergo a characteristic narrowing. This is a particularly valuable feature because these transitions are also allowed for paraexcitons [38]. A schematic picture of intraexcitonic transitions for the yellow series is presented in Fig. 2.7. The nominal transition energies of dipole allowed intraexcitonic and interexcitonic transitions for T=10K are shown in Table 2.1. 1s ortho- np energies are extracted from linear absorption and luminescence data, while 1s para – np energies are calculated by assuming a 1s ortho-para splitting of 12 meV, [39]. In addition to the intra-excitonic transitions in the yellow series, some of the yellow-green [40-42] and yellow-blue [43] transitions are also listed in the lower part of Table 2.1. Transitions to the indigo series are not included since they are higher in energy.

Fig. 2.7. Schematic drawing of intraxcitonic transitions.

Initial state Final state Series transition ∆E, meV 1s ortho 2p ortho y-y 115,6 1s ortho 3p ortho y-y 129,0 1s ortho ∞ y-y 139,1 1s para 2p para y-y 128,0 1s para 3p para y-y 140,5 1s para ∞ y-y 151,1 1s ortho 2p ortho y-y 235,0 1s para 2p para y-g 247,0 1s ortho 1s ortho y-b 547,0 1s para 2p para y-g 141,0

Table 2.1. Nominal transitions energies for inter- and intraexcitonic transitions at T=10K, [44].

Fig. 2.8. Induced-absorption spectrum due to the 1s-2p paraexciton transition obtained by Fourier spectroscopy technique. Cu 2O crystal of [100] orientation. Excitation energy 1,55 eV, T=4,2K, [43].

One of the first attempts to measure intraexcitonic transitions in cuprous oxide have been made by Goppert et. al. [45]. The Cu 2O sample was irradiated with a cw laser in order to create a population of n=1 excitons. The difference in the absorption spectrum of the irradiated sample with respect to a dark sample was measured with an infrared Fourier transform spectrometer. However, they did not succeed in observing a signal that might be attributed to the paraexcitons. More successful measurements using pulsed laser excitation

38 were reported out in [38, 43, 46]. For the experiment carried out in [43] a 120 fs, 1,55 eV pump pulse was used to excite the excitons. As expected, the presence of excitons causes an additional absorption which manifests itself as a decrease of the intensity of the transmitted light around 129 meV. The induced absorption spectrum, measured in [43], is presented in Fig. 2.8. In this experiment the repetition rate of the pump-laser in use was varied in order to estimate the lifetime of the 1s paraexcitons. It was found that the value is around 3 ms, which was in fact predicted by Jolk et al. [47]. However, as one can see from the Fig. 2.8, the width of the line is very narrow. Since the excitons will have some kinetic energy distribution at the measured temperatures, the induced absorption line should be much broader than the observed width of around 1 meV. This circumstance is giving a doubt to trust the presented result and it can be ruled out that the absorption line is due to an impurity related absorption at the same energy as the paraexciton 1s-2p transition. A different approach to detect the paraexcitons was taken in [21, 22]. In these experiments two colour time-resolved pump-probe spectroscopy was used to first excite the excitons by a high energy optical pulse in visible range and subsequently detect the intraexcitonic transitions with a mid-infrared pulse. Before turning to these kind of experiments we will first introduce the principles of pump-probe spectroscopy and give a description how to produce such pulses.

2.2.1.1. Principles of the ultrafast pump-probe experiment

In general, an experimental investigation aims to provide a data set with sufficient accuracy to ensure that all the relevant features of the system under investigation have been captured, [48]. In particular, time-resolved techniques focus on obtaining a high temporal resolution, which allows monitoring some of the fastest processes that occur in nature [49]. The limiting factor is the duration of the interaction between the investigated system and the measuring tool. For instance, observing a flying bullet that changes its position in less than a millisecond requires a photographic camera which can take a picture on this time scale. The trajectory of the bullet can be subsequently reconstructed from a series of successive snapshots. For example, typical motions on the molecular scale occur with speeds of about 1 kilometer per second, which implies that following movements of atoms over the length of a chemical bond (~10 -10 m) requires a shutter time of about one tenth of a picosecond (1 ps = 10 -12 sec) [50]. Similarly to photography, in spectroscopy a system is investigated by analyzing its interaction with light. When a good time resolution is desired, or in other words when the goal is to obtain a “snapshot” of the system time evolution, the light-matter interaction must be reduced to a very short period of time. The most efficient way to realize this is by using

39 a flash of light instead of a continuous source. The system is thus observed only for the duration of the light pulse. From a succession of such “snapshots”, the system’s dynamics that occur on a timescale slower than the duration of the light burst can be reconstructed. In general, for a large statistical ensemble in equilibrium, the average properties are not time dependent. Therefore, in order to observe dynamical processes, a non-equilibrium situation must be created initially. In other words, a sub-ensemble should be instantaneously “marked” so that its time evolution can be observed separately from the rest of the system. Fortunately, ultrashort laser pulses provide also a convenient means for an impulsive excitation of a system. In this way, an ensemble having certain initial conditions can be optically labeled and consequently its evolution can be followed separately from that of the rest of the system. This is exactly how the optical pump – optical probe experiment works (Fig. 2.9), which represents one of the most utilized techniques in time-resolved spectroscopy. The system under investigation is initially prepared in a non-equilibrium state by an interaction with a first laser pulse, called “pump”. After a certain delay, a second laser pulse, usually termed “probe”, investigates the current status of the system. By varying the delay between the pump and the probe, the evolution towards equilibrium is obtained in the form of a succession of snapshots. It is worth mentioning here that usually the system dynamics are not monitored in real time, as they occur to fast to allow varying the time delay and recording the data. Instead, a single snapshot is collected at a certain time delay, and then the system is allowed to reach equilibrium, so that a new interaction with the pump would yield the same effect. After that, a new snapshot is obtained by repeating the experiment for a different delay between pump and probe.

Figure 2.9 Schematic representation of the pump-probe setup.

To exemplify these ideas, let us consider a practical example, Cu 2O. As mentioned in the previous paragraph, it is possible to provide and detect the transition between the n=1 and other terms with higher principal quantum numbers. At time zero, an ultrashort pump pulse (Fig. 2.10), resonant with the orthoexciton energy, promotes a certain amount of particles into the 1s excited state. The orthoexcitons then decay to paraexcitons, via, for example, down-conversion processes discussed in paragraph 2.1. In time, relaxation will take place and the system will return to the equilibrium situation. During this time-window, the initially excited excitons can be interrogated by a second (probe) pulse, exciting them from the 1s to, for instance, the 2p excited level. The observed spectroscopic signal is the difference in transmission of probe pulses without paraexcitons (no pump) and with paraexcitons (pump). What kind of information can be obtained from this experiment? First of all, the lifetime of the 1s excited state is retrieved directly from the evolution of the pump-probe signal amplitude. Second, this would allow us to measure how fast the two types of excitons interchanged. For instance, the conversion time from 1s ortho to 1s para excited state. Additional information can be obtained by analysis of the transition lineshape, i.e. some statistical parameters like chemical potential and gas temperature.

Fig. 2.10. Schematic representation of time-resolved experiment on intraexcitonic transitions.

2.2.1.2. Experimental details

A schematic drawing of the pump-probe setup is shown in Fig. 2.11. The time difference between the pulses is provided by a mechanical computer-controlled delay stage. Both pump and probe pulses are focused on the sample to a diameter of 1mm and 0,3 mm respectively. The large spot size of the pump pulse ensures a better spatial overlap with the probe pulse and a more homogeneous probe region. The transmitted infrared light is collected using a commercial monochromator (Oriel) and then registered with a MCT detector cooled with liquid nitrogen. The pump and probe photons needs to be of certain energy: pump in the visible range (1,5-2,5 eV) and probe in mid-infrared (100-150 meV). To produce such photons, a widely adopted method is based on optical parametric generation that uses some non-linear medium. In optical parametric generation a photon of certain frequency is split in to two photons of lower frequencies. It is based on a second order nonlinear optical process - three photon parametric interaction in non-centrosymmetric crystals, [51]. According to energy conservation, the frequency matching condition that need to be satisfied is ω3=ω1+ω2, where the subscripts 3, 1 and 2 corresponds to the pump pulse, and two parametric waves, which are called the signal and the idler , respectively. The parametric light converters can be tuned over wide frequency range.

Fig. 2.11. Time-resolved pump-probe experimental setup.

Momentum conservation dictates that a pair of signal and idler waves can only be amplified efficiently when the phase-matching condition n3ω3=n 1ω1+n 2ω2 is satisfied, simultaneously with frequency matching condition. The frequency of the idler and the signal can be tuned

42 continuously through changing the indices n1, n2 and n3 by controlling the crystal orientation, it’s temperature, pressure or electric field applied to the crystal. The energy values of the signal and the idler are in the near-infrared region. In order to produce pulses in the visible range one needs to play with the signal and idler pulses: making their summation frequency or second harmonic generation. When one needs mid-infrared pulses around 100 meV, difference frequency generation can be used. This is, as well, a parametric process with the phase-matching conditions ωDF =ωS-ωI and k DF =k S-kI. In our experiment two commercial parametric amplifiers (Topas, Light Conversion Ltd.) were used to produce pump and probe pulses. The optical scheme of a TOPAS in the horizontal and in the vertical plane is presented in the Fig. 2.12. From the figure one can easily notice that the input beam passes the nonlinear crystal five times. In the first pass (A1-M1) the diameter of the input pump beam is reduced by telescope system and further focused by a cylindrical lens in order to generate superfluorescence in the nonlinear crystal. The iris aperture A1 controls the intensity of the pump beam. The beam splitters BS1 and BS2 couple the beam to the fourth and the fifth pass. In the second pass the weak superfluorescence signal is amplified by several orders. These two stages act as preamplifiers and shape the beam. In the third pass (BS2-M3-M3-M3’-M4-GP-M5-NC- DG-M7) a fresh pump-beam coming from BS2 passes through the nonlinear crystal and is incident on the grating. The grating selects the parametric signal in the fourth pass. The fifth pass starts from the beam splitter BS1 and forms the parametric seed beam for the power amplifier. This parametric seed beam is overlapped in space and time with the fresh pump beam on the mirror M11. A commercial Ti:Sapphire amplified femtosecond laser system (Hurricane, Spectraphysics) is used as an initial pulse source for both optical parametric amplifiers (Fig. 2.13). Typically the output of such system delivers output pulses of 120 fs (1 mJ per pulse) at 1 kHz repetition rate, with a center wavelength of 800 nm. The regenerative amplifier is designed to amplify individual pulses from a mode- locked Ti:Sapphire laser using a quasi-continuous high power Nd:YLF laser as pump source (Evolution, Spectraphysics). This laser system consists of a diode-pumped Nd:YLF laser head, optical resonator, acousto-optical Q-switch (© SpectraPhysics) and a LBO

(Lithium triborate) frequency doubling crystal. Nd:YLF (Nd:LiYF 4) is used as a working laser medium . This compound has two laser transitions at 1053 nm and 1047 nm of different polarizations. Usually the 1047 nm radiation is utilized due to a higher gain cross section. Such a system is capable of producing pulses with an average power of about 7,5 W at 523.5 nm at a repetition rate of 1 kHz. The Q-switcher (© SpectraPhysics) can be described as follows. A special designed piezo-electric transducer generates an ultrasonic

43 stationary wave in a transparent medium. The photo-elastically generated periodicity of high and low refractive indexes works as an optical phase grating.

Fig. 2.12. Schematic drawing of the optical parametric amplifier. A: iris aperture; Bs – beam splitter, L – lens, Cl – cylindrical lens, M – mirror, Cm – cylindrical mirror, NC – nonlinear crystal, TD – time delay, DG – diffraction grating, GP – path compensation glass plate.

Fig. 2.13. The schematic diagram of the laser system.

Before entering the regenerative amplifier, femtosecond pulses are stretched in order to avoid damage of the optics inside the amplifier cavity due to self-focusing. When a pulse is stretched its peak power is distributed in time and hence its power density is lowered. The stretcher is based on dispersion by diffraction gratings. The source of the femtosecond pulses (MaiTai, Spectraphysics) is the mode-locked Ti:Sapphire resonator pumped with a diode laser. The cw diode-pumped laser (Millenia, Spectraphysics) emits 532 nm light with an output power of about 5W. Since Ti:Sapphire has a broad absorption band in the blue and green region the output of the 532 nm cw-laser acts as an ideal pump. This system delivers continuously tunable pulsed output at 80 MHz repetition rate over a range of near infrared wavelengths from 790 - 810 nm with a pulse width of about 80 fs. The described system can be operated essentially as a ‘black box device’.

For these experiments described, we used a [100] platelet sample of Cu 2O with a thickness of 200 µm, cut from a crystal which has been grown by a floating zone technique. The samples was kindly provided by A.Revcolevschi (University of Paris IV) The sample was polished and mounted in a continuous flow cryostat (Cryovac) (base temperature 1,2 K).

45 2.2.2. Thermally-induced up-conversion

Here we describe another method of probing the 1s paraexciton excited state. This method is based on a thermally induced up-conversion process, partially described in section 2.1.

2.2.2.1. Principle of the experiment

As mentioned in section 2.1. there are a few processes which influence the ortho and paraexciton population. One process involves paraexcitons which are ‘converted’ in to orthoexcitons via a thermally induced spin-flip (Fig. 2.14). The energy gap of the exchange splitting is around 12 meV, which is in the order of 140K. If the gas temperature is low enough, at late times, when all ‘normal’ orthoexcitons already decayed, we can detect a weak luminescence from the orthoexcitons, which result from paraexciton up-conversion. Moreover, at high temperatures the particles will have high kinetic energy, increasing the probability of the up-conversion through an Auger process.

Fig. 2.14. Schematic representation of the time-resolved luminescence experiment. Since the orthoexciton 1s level is populated from paraexciton level, the detected luminescence reflects the amount of paraexcitons.

This experiment can be done in a time-resolved fashion. The temporal dynamics is achieved by triggering the excitation source and detection system. Once the system ‘knows’ when the sample was excited it can activate the detection system at any time after the researcher prefer. This technique has, not surprisingly, worse resolution than the pump-

46 probe experiment described in the proceeding paragraph, it is limited by the response time of the detection system, the shape of the excitation ‘pulse’ and the response time of the attendant electronics. But it is useful to measure a signal at long time up to ms with minimum resolution of few tens of ns. If one would like to provide such times in pump- probe experiments, one needs to construct a delay line with a distance from Groningen to Amsterdam. It is important to note that in such experiments we do not detect the paraexciton gas by itself. We can not estimate its temperature or other statistical parameters. Moreover, the lifetime of the excitons at 100K differs from the lifetime at 4K, due to the changes of the diffusion coefficient, for example. The below described experiments were done at Gonokami’s Group by M. Otter, N. Naka and M. Yoshioka at University of Tokyo [52] and are more fully described in [53].

2.2.2.2. Experimental details

The experimental setup is shown in Fig. 2.15. All measurements were done at a sample temperature of about 77 K. At this temperature the para-to-ortho conversion is efficient enough to see the luminescence (Fig. 2.4 part 4). At 77 K the transition rate is 10 8 higher than for the temperatures of 4 K. A continuous wave argon ion laser (5W, 514 nm, Millennia - Spectra Physics) is used as a pump for the mode-locked Ti-Sapphire system (Mira 900 laser, Spectra Physics). The optical cavity is specifically designed to utilize changes in the spatial profile of the beam

Fig. 2.15. Time-resolved luminescence experimental setup.

47 produced by self-focusing in the Ti:Sapphire crystal. The output pulses (200 fs, 800 nm, repetition rate of 76 MHz), are fed into a compact regenerative amplifier system (RegA 9000, Coherent Inc.). The diode laser (Verdi, Coherent Inc.) operated at 10 W is used as a pump source. The excitation pulse is focused on the sample by a lens (f=10 cm). The sample is placed in a continuous flow cryostat with a base temperature of 77 K. The emitted light is collected into a photomultiplier (R 636-10, Hamamatsu). A filter (XB106, Omega optical) cuts all the wavelengths except for the range 605 to 615 nm. The integrated luminescence intensity temporal dynamic, averaged over a few hundred pulse periods, is was recorded with a digital oscilloscope (Textronix TDS 744A). For the thermally-induced up-conversion experiments we used few [100] platelet samples of Cu 2O of different crystal qualities with a thickness of around 200 µm, cut from a crystal which has been grown by a floating zone technique. The samples was partially provided by A.Revcolevschi (University of Paris IV) and M.Gonokami (University of Tokyo).

48 2.2.3. Lifetime and discussion of exciton loss processes.

The transient changes of the transmission in the vicinity of the intra-excitonic transitions after one-photon excitation (E p=2.07eV) are presented in Fig. 2.16. The presence of excitons causes additional absorption bands, manifested as a decrease of the intensity of the transmitted probe beam near the intra-excitonic absorption energies. A summary of the dipole active intraexcitonic transitions is presented in Table 2.1.

Fig. 2.16. Transient induced transmission spectrum originating from the presence of

excitons in Cu 2O ([100] crystallographic orientation). Excitation energy 2,1 eV, bath temperature T = 10 K.

The spectral width of the probe pulse is around 30 meV. As a result, one may expect to observe several intraexcitonic transitions within the probe energy range. As a matter of fact, the probe pulse spectrally covers 1s-2p orthoexciton transition, 1s-2p paraexciton and 1s-3p orthoexciton transitions (Fig. 2.16, Table 2.1). The energies of 1s-2p paraexciton and 1s-3p orthoexciton transitions are lying close to each other ( ∆=1 meV). As was mentioned in §2.2.2.2., the spectral resolution of our detection system is about 0,3 meV. However, we can not resolve these two lines, since each line is thermally broadened: the width of the line at T=8K is around 1,6 meV, [39] (see also §3.3). The 1s-3p paraexciton transition line would definitely be preferable for investigation. However, there is an additional strong absorption band at those frequencies (Fig. 2.17), making experiments in this energy range more difficult.

Fig. 2.17. Infrared absorption band in Cu 2O in [100] direction (T=10K).

The origin of this band is a topic of discussion. In [54-57] it was assigned to a transition between two different sets of excitons, i.e. between yellow and green exciton series. In [58] this band was assigned to a fundamental lattice vibration, though no indications of such a high-frequency fundamental vibration are available from inelastic neutron scattering studies [59] and lattice dynamics calculations [60]. Another possible explanation was given in [61], where it was assigned to silicon impurity states. In [62] it was argued that this band is due to a multiphonon process. However, in [63] a comparative study of the temperature dependencies of the peak frequency and the band width on one hand and those of infrared and Raman active fundamental vibrations on the other hand showed the one-particle nature of this band. Finally, in [64] it was proposed that the 144 meV feature originates from a bound bi-phonon state split off from the two-phonon band ω(k)+ ω(-k) . Independent of the origin of the 144 meV band, it is obvious that the study on paraexcitons from 1s-3p transition is hampered by its presence. We therefore focus on lower energy para 1s-2p transition. Fig. 2.18 shows a few shots from the temporal dynamics of intraexcitonic transition lines at T=8K. First, the 1s-2p orthoexciton transition near 115 meV builds up in a few tens of picoseconds. After this, the 1s-2p ortho-

Fig. 2.18. Induced changes in the intra-exciton transmission spectrum at various times after excitation by one-photon absorption at 2.07 eV (~700 µJ/cm2). The thick black solid

lines schematically indicate the paraexciton part. T=8 K. Cu 2O of [100] orientation (provided by A. Revcolevschi, University of Paris IV).

Fig. 2.19. Shift of the center of mass of the 1s-2p para / 1s-3p ortho transition line as a function of time due to the orthoexciton decay.

51 para

ortho

Fig. 2.20. Time dynamics of spectral changes induced with one-photon excitation at 2.07 eV and T=8K. The result was obtained by surveying ∆T/T values at certain parts of the spectrum (gate window width 0,5 meV): circles – 128 meV line (1s-2p para), triangles – 129 meV line (1s-3p ortho), grey line – 115,6 meV line (1s-2p ortho). Inset: 1s-2p

paraexciton transition (semi-logarithmic scale). Cu2O of [100] orientation (provided by A. Revcolevschi, University of Paris IV). exciton band decays exponentially with a time constant of about 1.5 ns. This is indeed what one expects for the orthoexcitons, based on numerous fluorescence studies, [20, 65, 66] (and references therein). In the vicinity of the paraexciton 1s-2p transition, we also observe an induced absorption band with a similar build-up time. As time evolves, the centre of mass of this band shifts to lower energy (Fig. 2.19). This shifting saturates after about 1,5 ns. The origin of this shift lies in the near coincidence of the ortho 1s-3p and para 1s-2p energy levels (see Table 4.1). Also there is a shift of the paraexciton line itself; discussed in detail in §2.2.4 The ortho 1s-3p band has the same time dynamics as the 1s-2p ortho transition at 115 meV. As it was discussed above, the formation of the paraexciton occurs through an ortho-para down conversion process, and thus takes a longer time to

52 build up (general overview in Fig. 2.20). The observed time constant for the down conversion process is about 300 ps [21], which is consistent with earlier experiments, [38]. Since the 1s-2p paraexciton energy is slightly below the 1s-3p ortho energy, these two effects lead to a gradual shift of the observed band. Once the orthoexciton has decayed, the induced absorption around 130 meV is solely due to the 1s-2p para transitions. The overview of the observed early time dynamics is shown in Fig. 2.20. From the inset in Fig. 2.20, it is clear that paraexciton decay has a double exponential behavior. The initial fast exponential decay ( τ1=1,5 ns) is partly due to small contributions of the nearby 1s-3p orthoexcitonic transition and partly due to the up-conversion of hot paraexcitons back to the 1s orthoexciton state, which may subsequently decay radiatively. The time constant for the decay of the para-exciton absorption after 1.5 ns can not be reliably determined from the present data, the only indication the data gives is that the lifetime is at best in order of a few tens of ns. This observation is contradicting expectations. As discussed above, the paraexciton lifetime should be extremely long (not ns), since the transition is forbidden in all orders of perturbation. Let us now discuss it somewhat more detailed.

2.2.4. Mechanisms of paraexciton losses.

The processes which influence the orthoexciton population are mentioned in paragraph 2.1.2 (Fig. 2.4). The processes for paraexciton decay are also partially discussed there. As one can see, some of the mechanisms for orthoexcitons are applicable for paraexcitons. Fig. 2.21 depicts the most important decay processes for paraexcitons: thermally induced up- conversion (panel 1), Auger recombination (panel 2) and paraexciton trapping by the crystal imperfections (panel 3). One of the intrinsic decay processes is the thermally induced up-conversion of the paraexciton to the orthoexciton (Fig. 2.21, panel 1). The transition probability of such conversion is strongly temperature dependent and may be expressed by:

 ∆   −   k BT  ρP→O ∝ e , )9.4( where ∆ is the energy difference between the ortho and paraexciton states (12 meV). At elevated temperatures one can use this decay channel to indirectly observe the para-exciton decay by monitoring the luminescence from the upconverted ortho-excitons [53], as discussed in §2.2.2. At low temperatures, however, the transition probability is so low that this process can be neglected.

53 The Auger processes mentioned in §2.1.2 for the orthoexcitons could also limit the lifetime of the paraexcitons. From a study of the phonon-assisted emission of orthoexcitons, O’Hara et al. [6] came to the conclusion that the coefficient of Auger recombination process for ortho-excitons is as large as A=7*10 -17 cm 3/ns.

Fig. 2.21. Schematic diagram of ortho and paraexciton state interaction – possible processes of paraexciton losses. 1 – thermally induced up-conversion; 2 – Auger recombination, 3 – trapping by impurities.

54 This channel should be equally effective for paraexcitons. The value reported by Wolfe et al. [66, 67] was about 10 -16 cm 3/ns. On the other hand, Jolk et al. [47] have claimed, based on a pump-probe measurements involving the higher Rydberg series in the visible region, that the Auger coefficient is smaller by several orders of magnitude. Recent calculations [11] have also shown that the Auger process recombination rate should indeed be negligible given the long paraexciton lifetime. In general, the Auger recombination of paraexcitons should be highly dependent on the density of the excitons as well as any decay process involving two particles. As an example, in Fig. 2.22, the time decay of the paraexcitons due to the is shown for different pump energy densities. Since the initial density of the excitons is different for different powers, one may consequently expect a higher contribution from the Auger process for the highest pump densities. However, the decay curves do not depend on the pump power density, showing that Auger processes are indeed irrelevant for the pump power densities used.

Excitation energy density 0,05 J/cm 2 0,1 J/cm 2 0.25 J/cm 2 0,5 J/cm 2 0,75 J/cm 2

1 J/cm 2

Normalized intensity, arb. units arb. intensity, Normalized 0 5 10 15 20 25 30 35 40 45 50 Time, ns

Fig. 2.22. Decay curve of the paraexciton population for different excitation powers. Results obtained from time-resolved luminescence experiment at 77K on the sample with high impurity concentration. Excitation wavelength 800 nm, Cu 2O of [100] orientation. The sample and the experiments were provided by M. Otter, N. Naka, K. Yoshioko, M. Gonokami at University of Tokyo, [52, 53] .

55 There is another process, which is not presented in Fig. 4.21, but which might be important. Since the sample has a different symmetry near the surface one can expect additional mechanisms of exciton loss. It is possible to check this experimentally. One may carry out the described above pump-probe experiment on 1s-2p paraexciton transition using for excitation photons of different energies. Since photons with different energies have a different penetration depth in the sample, one can vary the relative contribution from the surface. For example, the creation of the yellow electron –hole pair (C 1-V1) by one-photon absorption (2,07 eV) leads to creation of excitons in the vicinity of the surface (penetration depth ~30 µm, [Katarzyna]). In contrast, two-photon absorption using 1,55 eV photons (penetration depth ~1 mm) primarily excites excitons in the bulk, [65]. In this latter case blue electron-hole pairs (C 1-V2) are initially created with subsequent relaxation to the yellow series. Fig. 2.23a shows the induced absorption spectra of intraexcitonic transitions at different times after creation of the exciton gas using two-photon excitation. The normalized early time dynamics of the 1s-2p transition for both excitation methods are shown in Fig. 2.23b. The average excitation power was the same in both of the experiments.

Fig. 2.23. (a) Induced changes in probe spectrum versus time using two-photon absorption excitation method with 1.55 eV (~200 µJ/cm 2), T=8K. (b) Early time dynamics of 1s-2p paraexciton transition. One-photon excitation at 2,07 eV – open circles, two-photon excitation with 1,55 eV – filled circles. T=8K, pump density ~200 µJ/cm 2 in both

experiments. Cu 2O of [100] orientation (provided by A. Revcolevschi, University of Paris IV).

56 Since there is no difference in decay time constant for the gas created near the surface and in the bulk one can conclude that there is no additional observable process which contributes to the exciton losses due to the presence of the surface. Furthermore, the presented result (Fig. 2.23b) again shows that Auger recombination does not play a significant role. The volume, where the exciton gas is created is different for discussed excitation methods. As it will be shown later on in §2.2.4, with one-photon absorption (2,07 eV photons) one may achieve densities of around 10 18 cm -3, while with two-photon absorption (2x1,55 eV) the density of the 1s excited paraexcitons is around 10 17 cm -3. Since the densities differ by an order of magnitude, one would expect a faster decay in the one-photon excitation case when Auger process would be important. This is clearly not observed. In Fig.2.21 (panel 3) one more mechanism is presented which is based on the exciton trapping by crystal imperfections. Since the excitons diffuse in the crystal, there is some finite probability for them to be trapped by impurities. This process can definitely shorten the lifetime and, in particular for Cu 2O, it is extremely important since the paraexcitons exhibit a surprisingly high diffusion constant. Trauericht et al. [68] investigated the paraexciton gas in a potential trap and observed a tremendously high diffusivity. The value of diffusivity was much higher than it was predicted by standard deformation-potential analysis. As an illustration of the extreme diffusivity, Fig. 2.24 shows two spatial and time- resolved images of exciton luminescence, taken from the work by Trauernicht et al. [61]. The expansion of the cross-sectional area of the paraexciton cloud corresponds to a diffusion constant of 600 cm 2/s at 2K.

Fig. 2.24. Time-resolved images of the paraexciton luminescence in Cu 2O at T=2K. The time difference between a) and b) is 400 ns. The dashed lines represents the spatial resolution. (Figure reproduced from [68] with permission).

57 In order to investigate the influence of defects and impurities on the paraexciton lifetime we carried out time-resolved experiments (§2.2.2) for two samples of different qualities, named L sample (the sample was provided by M. Gonokami’s group, University of Tokyo) and S (the sample was provided by A. Revcolevschi’s group, University of Paris IV) sample for simplicity. Also luminescence spectra for L- and S-samples were measured: as it was mentioned in Chapter 1, the amount of impurities, i.e. sample quality, can be evaluated from photoluminescence experiments, [69]. The exact spectral structure and assignments of the lines lying inside band-gap are discussed in §3.3. The luminescence spectra for samples L and S are shown in Fig. 2.23a1 and Fig. 2.23b1. It is clear, that the paraexciton lifetime created in sample L is much longer (2 µs) than for the particles created in sample S (10 ns), (Fig. 2.25 a2,b2).

Wavelength, nm Time, ns 600 700 800 900 1000 0 20 40

Cu -

sample S

O O+1 I

Exciton n

t e

i t

Exciton y Intensity O+2

O+1 sample L

Cu -

600 700 800 900 1000 0 2000 4000 6000 Wavelength, nm Time, ns

Fig. 2.25. Left panels : Luminescence spectra measured under identical experimental conditions for S–sample (a1) (sample was provided by A.Revcolevschi’s group, University of Paris) and L-sample (b1) (sample was provided by M.Gonokami’s Group, University of Tokyo). Excitation wavelength 532 nm, T=77K. Right panels : Paraexciton population time decay for S-sample (a2) and L-sample (b2) at 77K, measured by time-resolved luminescence technique. Excitation wavelength 800 nm. The experiments was carried out at University of Tokyo by M.Otter, N.Naka, K. Yoshioko, M.Gonokami, [52, 53].

The amount of the impurities can be qualitatively evaluated from the comparison of the impurity line strength with respect to the exciton luminescence line. Form panel a1 and b1 of Fig. 2.25 one may easily see, that samples L and S are different in the amount of oxygen and copper vacancies. Fig. 2.25 shows that the ratio of the oxygen peak intensity to the exciton peak intensity for both oxygen peaks increases by about one order of magnitude when the lifetime decreases from the order of µs to ns, indicating that the oxygen vacancy concentration might influence the lifetime. More strikingly, however, is the difference in Cu vacancy concentration. For sample S, where the paraexciton lifetime is short, the ratio of copper vacancy intensity to exciton intensity is in order of 10 4, whereas for sample L the ratio is in order of 10 -1. The correlation of this ratio with the lifetimes strongly suggests that the main mechanism for paraexciton loss is trapping by Cu vacancy.

2.2.4. Exciton trapping by the crystal imperfections.

Exciton trapping by impurities in solids have been investigated theoretically and experimentally by several authors [12, 35, 60, 70-81]. Some models for capture into deep levels were proposed, namely multiphonon capture [82], cascade capture [83, 84] or Auger capture of free carriers [85-89]. None of these mechanisms can account for the experimental facts in Cu 2O, [90]. Another concept, Auger recombination of exciton via a deep impurity level, was proposed in [90]. In this model the capture of an electron into a deep level may occur when a free exciton meets the impurity. The electron from the exciton is captured by the impurity, and the excess energy is transferred to the hole which is thereby highly excited into the valence band. Processes like this are known in scattering theory as rearrangement collisions . Such a capture of an electron by an impurity is quite similar to the scattering of a proton by a hydrogen atom where the electron is picked up by the incident proton. Analogously, a hole may be captured into a deep level thereby exciting the electron into the conduction band. Using the general conclusion from [90], the coefficient for electron or hole capture depends on the bands from which the impurity wave function is mostly composed.

In general, impurities in Cu 2O, as well as in other semiconductors, can form either charged or neutral potential traps. The excitonic capture, theoretically studied in [90] is based on the idea that when a free exciton approaches an impurity, one of the electronic particles of the exciton is attracted by the ionized impurity. If this particle (electron or hole) is captured by the impurity, the excess energy is transferred to the other particle of the free exciton, which then is excited into the continuum in an Auger-like process. This basic

59 process is illustrated in Fig. 2.26a for the case of a free exciton interacting with an ionized impurity. In our case the ionized acceptor is the vacancy state, formed by taking a copper ion Cu + out from the crystal (see also Chapter 1). Since this vacancy carries a negative charge, the hole of the free exciton can be attracted.

Fig. 2.26. a) Illustration of the exciton trapping by the ionized impurity. b) Interactions between the particles participating in the hole capture process.

The Hamiltonian for such a system can be split into a steady-state part and a perturbation for the initial (free exciton and charged impurity) and for the final state (bound hole to the impurity and hot electron) of the process (Fig. 2.26b). The transition probability from the initial to the final state can be written using Fermi’s golden rule:

2π 2 W = d 3k M (k) δ (E − E )D(k), (2.10) h ∫ i f

60 Here M( k) is the transition matrix element, D( k) is the density of states in k space of the final state. The δ function between the initial energy Ei and the final energy Ef enforces energy conservation. Assuming parabolic energy bands and an isotopic matrix element one can obtain:

2mk 2 W = M (k) , (2.11) πh3

The complete Hamiltonian that describes the electron, the hole, and the impurity contains a background term H 0 plus the three interaction terms shown as well schematically in Fig. 4.26b:

H = (H 0 +U E ) + (V1 +V2 ), (2.12)

In our calculation the hole-impurity interaction has been considered as a perturbation causing the transition, meaning that we have to calculate the matrix element of V 2. The transition matrix element in the Born approximation may be written as follow:

K κ M = xi V1 yk , (2.13)

K κ where |x i > is the eigenstate (H 0+U E) of the free exciton and |y k > is the eigenstate of the hole bound to the impurity plus the highly excited electron (H 0+V 1). The next step, of K κ course is to give the explicit wave functions for the | xi > and |y k >. The eigenstates of a free exciton with a momentum K are expanded in terms of eigenstates of the crystal with a lattice of L sites [94]:

xK =ψ i (r ,r )u (r )u (r ) = i 1 2 c,k1 1 v,k2 2

1 i iβr1 i(K −β )r2 = 2/1 ∑ AK ,β e e uc,β (r1)uv,K −β (r2 ), 14.2( ) L β

Here uc,k (r 1) and uv,k (r 2) are the periodic parts of the electron and hole Bloch functions, i i and A K, β is the Fourier transform of the internal wave function ψ (r 1,r 2) of the exciton. The κ final state |y k > is described by a product of the wave functions of the hole bound to the impurity and that of a highly excited electron. In general, the wave function of a hole bound to the impurity is again a superposition of Bloch functions [95];

61 j 1 j ikr 1 φt (r1) = /1 2 ∑Cn,ke un,k (r1), 15.2( ) L n,k

whereas the highly excited electron with momentum κ is described by:

iκr2 φ(r2 ) = e uc,κ (r2 ), 16.2( )

Finally, the interaction potential between the particles are of the form (screened Coulomb interaction): e2 V (r) = e−k s r , 17.2( ) εr

In order to take into account many-particle effects in a simple approximation, here the inverse screening length ks was added since there is a screening of the Coulomb interaction. Now we have all the prerequisites to calculate the transition matrix element M. By inserting the wave functions from Eq. 2.14, 2.15, 2.16 as well as the interaction (Eq. 2.17) into the Eq. 2.13 for the matrix element we get for the leading term:

1   3 3  * −iβ 'r1 *  −iκr2 * M = ∫ d r1d r2 ∑Cn,β 'e unβ ' (r1)e ucκ (r2 ) ⋅ L  n,β ' 

e2e−k S r2    iβr1 i(K −β )r2  ⋅ ∑ AK ,β e uvβ (r1)e uc,K −β (r2 ), 18.2( ) εr2  β 

Now we exchange the integrations over r and β, getting:

1 * 3 3 −iβ 'r1 * −iκr2 * M = ∑∑Cn,β ' AK ,β ∫ d r1d r2e unβ ' (r1)e ucκ (r2 )⋅ L n,β' β

2 −kS r2 e e iβr1 i(K−β )r2 ⋅ e uvβ (r1)e uc,K−β (r2 ), 19.2( ) εr2

62 First, the integral over r 1 can be executed immediately, yielding:

1 3 * −iκr2 * M = ∑∑∫ d r2Cn,β ' AK ,βδnv δ ββ 'e ucκ (r2 ) ⋅ L n,β' β

2 −k S r2 e e i(K −β )r2 uvβ (r1)e uc,K −β (r2 ), 20.2( ) εr2

In order to calculate the remaining integral over r2, the Bloch factors uck are expressed by a Fourier series:

iQr u ck (r) = ∑ BcQ (k )e , 21.2( ) Q

Now allowing for the calculation of the final integral I over r 2:

2 * 4πe 1 I = ∑ BcQ (κ)BcQ ' (K − β ) 2 2 , 21.2( ) Q,Q' ε kS + (K − β − κ − Q − Q )'

The main contribution to this sum comes from the first Brillouin zone [96], i.e. Q=Q’=0, getting:

2 K κ 4πe AK ,β Fvv Cc,K−β M = xi V1 yk = − ∑ 2 2 , (2.22) εL β kS + (K −κ − β)

Here F vv is the overlap integral, which can be written as: 1 F = d 3ru * (r)u (r ,) (2.23) vv L ∫ cκ v,K−β

For the case of a ground exciton state (1s) some simplifications can be made. For instance, the overlap integrals containing functions of the same band are equal to unity

63 Fvv =δvv , whereas those with functions of different bands vanish, [97]. Furthermore, for isotropic parabolic bands the exciton wave function in real space may be written as:

3 2/1 1s iKR  a  −aρ ψ (R,ρ) = e   e , (2.24)  π  where R is the center of mass coordinate, ρ is the coordinate of relative motion and a is the reciprocal excitonic Bohr radius. The most difficult problem remaining now is to get some realistic simple approximation for the wave function of the hole bound to the impurity. Since the a priori calculation of the impurity wave function is beyond the scope of the present investigations, we will use one of the models frequently employed in literature, [95, 98-101]. It appears to be the fact that for the impurities there is no direct relation between the energetic position of the impurity level and the spatial localization of the impurity wave function [90]. For instance, we can use the effective-mass and δ-potential model of Lucovsky [102], meaning that the localized impurity potential is approximated by a δ-function. The respective wave function with the localization parameter γ for δ-potential [102] and effective-mass 1s hydrogen-like wave function [87, 88] read as:

 γ  2/1 ψ 1s (r) =   e−γr  2π   γ  2/1 1 ψ δ (r) =   e−γr , (2.25)  2π  r

The Fourier coefficients of the 1s exciton wave function and the impurity wave function are given by, [87, 88, 102]:

 3  2/1 α  8πα AK ,β =   2  π  ()α 2 + ()K − β 2  γ  2/1 4π C1s (β) =   , (2.26)  2π  γ 2 + β 2

64 Some theoretical estimations of the capture coefficient for a 1s excited state are already done in literature [87, 88, 90, 95, 98-100, 102]: using the effective-mass impurity wave function with localization from 0,5 to 4 lattice constants, the absolute value of the capture coefficient varies between 10 -18 cm 3/ns and 10 -14 cm 3/ns. We are not going to repeat here this long and difficult calculation, but we do want to point out some of the conclusions arising from them. The transition probability happens to be strongly dependent on the localization parameter and the excitonic Bohr radius:

1 W ∝ 3 3 , (2.26) γ a0

From qualitative considerations one expects that stronger localization causes a larger extension of the wavefunctions in k space, which in turn should lead to an increased transition probability. In contrast, from Eq. 2.26 it follows the transition probability decreases with increasing localization. The reason for this surprising result is that for the scattering process the localization of the electron and hole within the exciton also plays an important role. The transition matrix element (Eq. 2.21) contains a product of the Fourier transforms AK, β and Cvβ of the hole and impurity wave functions, whose shape is governed by the least localized one of the wave functions (the exciton). The impurity wave function, which is more localized and has less variation in k space, merely enters with its amplitude at k=0 which decreases with increasing localization. Next conclusion is that the transition probability increases with the decrease of the excitonic Bohr radius (Eq. 2.26). Therefore, large capture coefficients are expected in a case of 1s paraexcitons in view of their extremely small free exciton Bohr radius of 0.7 nm. The model we have discussed above contains the idea of a “single-shot” trap: once it traps the exciton it is not able to trap another one. In this case, we should be able to “switch-off” all impurities by the certain amount of excitons. In other words, if the exciton density is higher than some “critical” value, which depends on the impurity concentration and on the quenching probability, the lifetime should increase. This actually strongly contradicts with the experimental results on L sample (the sample was provided by M.Gonokami’s group from University of Tokyo) presented in Fig. 2.27. For the simple calculation of the exciton gas density we can assume, that the probability of the absorption of the photon and the consequent creation of the exciton is P=1. For the pump energy densities used in the present experiment (Fig.2.27) we can achieve an exciton gas density up to ~10 19 cm -3, almost 10 4 timeshigher, than an astimated impurity concentration (see also Fig. 2.29). Apparently, we are not able to saturate the impurities.

In order to explain the experimental results, we here propose a chain model of exciton trapping Fig. 2.28. The initial state of this process can be imagined as an ionized vacancy - VCu and a free exciton (Fig. 2.28a). Then a hole from the exciton will be captured by the ionized vacancy and the excess energy is transferred to the electron, which is excited into the continuum (Fig. 2.28b). Then, few processes can take place. For example, the acceptor hole and an excited electron form a bound exciton state (Fig. 2.28c). Recently, bound excitons were observed by Jang et al., [103]. Alternatively, recombination with another exciton at the same acceptor can take place Fig. 2.28e. This process can be described as a two-hole transition, [104]. Also, after process b, the impurity can be again ionized upon simple recombination of an electron and hole (Fig. 2.28d). The copper vacancy, while it is still in an ionized state, can after all attract a second free exciton and repeat the process (Fig. 2.28a). Thus, once the procedure is initialized, it can continue in a kind of chain reaction (Fig.2.28).

Excitation energy density 0,05 J/cm 2 0,1 J/cm 2 0.25 J/cm 2 0,5 J/cm 2 0,75 J/cm 2

1 J/cm 2

Normalized intensity, arb. units arb. intensity, Normalized 0 5 10 15 20 25 30 35 40 45 50 Time, ns

Fig. 2.27. Paraexciton lifetime at 77K for different excitation powers. Results obtained from time-resolved luminescence experiment on the sample with high impurity

concentration. Excitation wavelength 800 nm, Cu 2O of [100] orientation.The sample and experiments were provided by M. Otter, N.Naka, K.Yoshioka, M.Gonokami at University of Tokyo, [52, 53].

Fig. 2.28. Illustration of the “chain reaction” exciton trapping.

67 For the present model of highly mobile excitons in a system containing traps, we can estimate the lifetime of the free excitons as a function of the impurity density, assuming the diffusion coefficient is a known parameter. Random walks of a particle on a lattice that includes traps have been studied in [71, 75, 105-107]. Also for paraexcitons in Cu 2O some simulations are already done, [39], where the values of the diffusion coefficient vary from 0,5 to 20 cm 2/s. We also calculated the survival probability of an exciton diffusing with a coefficient D in a matrix of quenching sites. The distance between these quenching sites is approximated by a Poisson distribution with average distance L. In the limit of a high density or at long times the temporal dynamics can be described by, [108]:

1  t  2  t β   t  I(t) = I )0(   exp − exp  − , (2.28a)    β    τ diff   τ diff   τ 0 

2 3 2 −1 τ diff = []2π )2/3( Dn 1D , (2.28b)

1/3 Here τdiff is a characteristic diffusion time and n1D = (n 3D ) or n1D =1/L is the one- dimensional density of quenching centers. The first term is related to the movement of a particle in the matrix of quenching sites as was calculated by Balagurov et al. , [108]. The exponent β comes from the dimensionality of the process and equals 1/3, 1/2 and 3/5 for diffusion in 1D, 2D and 3D, respectively. The time dependence of the exciton lifetime τ0 accounts for the decrease in intensity when no trapping occurs. Eq. 2.27 is correct only for times which fulfill the condition: L2 t >> , (2.29) π 2 D

It is not difficult to understand this condition from the physical point of view: the time should be bigger than the average time needed for an exciton to travel the average distance between traps. The examples of fitting the experimental data with the described function (Eq. 2.28) are presented in Fig. 2.29 for samples S (samples was provided by A. Revcolevschi, University of Paris IV) and L (sample was provided by M. Gonokami, University of Tokyo) (see also Fig. 2.25). The fitting parameter was one-dimensional density of quenching centers n1D , while the diffusion coefficient was chosen being a fixed parameter D=0,5 cm 2/s. As a result, for sample S we suggested the impurity concentration of 10 15 cm -3 and for sample L – 10 11 cm -3. Also, from these fits, we can conclude, that at long time, the

68 lifetime behavior can be well described using a single exponential function, while at early times the non-exponential part ~t 1/2 (Eq. 2.28) contributes.

Intensity sample S sample L

Intensity

0 20 40 0 2000 4000 6000 Time, ns Time, ns Fig. 2.29. Fitting of paraexciton population time decay with Eq. 2.28: (a) sample S (sample was provided by A. Revcolevschi, University of Paris IV), (b) sample L (sample was provided by M. Gonokami, University of Tokyo). Experimental data was obtained at T=77K, measured by time-resolved luminescence technique at University of Tokyo by M.Otter, N.Naka, K. Yoshioka, M.Gonokami, [52, 53]. Excitation wavelength 800 nm. One- 5 -1 4 - dimensional impurity density n 1D was a fitting parameter (10 cm for sample S and 10 cm 1 for sample L). 1000 10 15 100 14 ) 10 I -3 10 cu

13 2+ 10 /I

1 exciton 10 12 0,1 10 11

density(cm 0,01 10 10 0 500 1000 1500 2000 2500 3000 Lifetime, ns Fig. 2.30. Numerical simulation of exciton lifetime using Eq. 2.28. Simulation 2 parameters: D=0,5 cm /s, τ0=3 ms. Black curve – numerical simulation; dots was obtained from the calculation of I Cu /I Exciton ratio for the S samples (close dots, samples was provided by A.Revcolevschi, University of Paris) and L samples (open dots, samples was provided by M.Gonokami, University of Tokyo).

For samples of different quality it has been measured the lifetime of the paraexciton gas and the impurity luminescence spectra by M. Otter, N. Naka, K. Yoshioka, M.

Gonokami at University of Tokyo, [52, 53]. By looking at the ratio of I Cu -/I exc for each vacancy spectrum measured we can relate this ratio to a vacancy concentration. Here we assume that the luminescence intensity linearly depends on the amount of impurities. This is shown in Fig. 2.30. The simulation fits the measured ratio quite well. Evidently, the copper vacancy concentration is directly related to the exciton lifetime.

2.2.5. Time evolution of paraexciton gas statistical parameters

As discussed in Chapter 1 and §2.1 luminescence spectra can give direct information about the statistical exciton gas parameters, i.e. temperature, chemical potential and density. As one can already guess, the statistical parameters of the exciton gas can not be determined through the time-resolved luminescence experiment via the orthoexciton state (§2.2.2). However, the pump-probe experiment of intraexcitonic transition is quite a powerful tool. As we will show in the following, the line shape and strength of the induced absorption (Fig.2.16) depends crucially on the statistical parameters. First, we will focus on a line shape analysis and will try to extract from the experimental results the chemical potential and the temperature of the exciton gas. Then we will connect analytically the strength of the absorption transition line and the gas density. Consider two dispersion curves for the 1s and np states as depicted in Fig. 2.31. Since the effective exciton mass in the 1s state is larger than in the np states ( m1s >m np ) the energy gap E1s-np (k) between this levels increases with k:

2 2   h k  1 1  E1s−np (k) = Ek =0 +  − , (2.29) 2  mnp m1s 

The transition rate W 1s-np can directly be written by applying Fermi’s Golden Rule:

r r r r 2π 2 3 W1s−np = µi f1s (k [1) + fnp (k )]δ [E1s−np (k ) − E]d k, (2.30) h ∫

70 Here µi is the transition matrix element, f1s , fnp , are the distribution functions of the 1s excitons, np excitons and E is an energy of the incident photon. From this point we need to make some assumptions. First of all we suppose that the transition matrix element µi is k- independent. Second, since the transitions in a cubic lattice are direction independent we can write: r d 3k = 4πk 2dk , (2.31)

Finally, we assume that the np levels are not occupied at all. In this case we can derive the line shape of the induced absorption as:

2 αind (E) ∝ ∫ k f1s (k)δ (E1s−np − E)dk , (2.32)

Eq. 2.32 can be transferred into an integral over E1s-np by using Eq. 2.29:

αind (E) ∝ ∫ f1s (k) E1s−np − Ek =0δ (E1s−np − E)dE 1s−np , (2.33)

h2k 2 Enp = 2mnp

h2k 2 E1s = 2m1s

Fig. 2.31. Schematic drawing of the dispersion of the generalized 1s and np states.

71 For a small gas density we can use the Boltzmann approximation for the thermal distribution within the 1s band. Such an approach is used in [39, 109]. Higher exciton densities will require the use of the Bose distribution function. Qualitatively, however it is clear that when the chemical potential approaches zero, the line shape will increasingly narrow, [37]. Finally, the analytical solution for the induced absorption line shape using the Bose distribution (Chapter 1) reads as:

E − E α (E) ∝ 1s−np k =0 , (4.34) ind E1s−np − Ek=0 µ −   k T  m1s  B  −1k BT e mnp  −1

As already clear from Fig. 2.31, the line shape should broaden with increasing temperature as well as with an increasing effective mass ratio. When the masses are equal the lineshape reduces to a δ-function at E=E k=0 . In the last step before presenting experimental data, homogeneous line broadening is discussed. This additional broadening can be neglected in the case of a 1s state since the linewidth is already in order of kBT. In order to estimate the homogeneous broadening of para np levels, we can compare to the broadening, observed in absorption spectra of the corresponding orthoexciton levels (Chapter 1, Fig. 2.4). There is no particular reason why it should be different for the paraexciton band, since the dephasing mechanism is phonon scattering which is not sensitive to the spin. Based on it, the final Eq. 4.34 needs to be convoluted with the usual thermal broadening Lorentzian function. Fig. 2.32 shows a number of spectra of the induced 1s-2p paraexciton absorption at different bath temperatures (left panels) and at different times after the excitation pulse

(right panels, bath temperature T=1,3 K) for Cu 2O crystal in [100] orientation, provided by A. Revcolevschi (University of Paris IV). These spectra were measured after two-photon absorption at a fixed time delay of 3 ns after excitation. A convolution of a Lorentzian thermal broadening with Eq. 2.34 was fitted to the data, where the fitting parameters are the transition line position Ek=0 , temperature T and chemical potential µ. We start with the analysis of the temperature dependence, presented at Fig. 2.32a. First of all, the transition shifts toward higher energy with increasing temperature (Fig. 2.33). Apparently, the involved levels do not shift with the bath temperature in a similar fashion. The origin of this is the decrease in the binding energy of the 1s level. Whereas both levels shift as a result of the reduction of the band-gap, the 1s level experiences an additional shift due to a reduction of the central cell correction energy.

a b

0.125 0.130 0.135 0.125 0.130 0.135

Bath T=45 K t=300ps Gas T=50 K Gas T=39 K dT/T

Bath T=35 K t=700ps Gas T=41 K Gas T=30 K

intensityNormalized

Bath T=30K t=1200ps Gas T=25 K Gas T=20 K

Bath T=10 K t=2000ps Normalized Normalized intensity Gas T=15 K Gas T=12 K

Bath T=1,3 K t=3000 ps Gas T=7 K G as T=7 K

0.125 0.130 0.135 0.125 0.130 0.135 Energy, meV Energy, meV

Fig. 2.32. Normalized induced absorption due to the 1s-2p paraexciton transition. Excitation via two-photon absorption using photons of 1,55 eV energy (2 mJ/cm 2). The black line is a fitting function –which is the result of the convolution of Eq. 2.34 (light grey) and the Lorentz broadening function (grey). The left plots (a) represents the results at different bath temperatures but fixed delay time at 3 ns. Right plots (b) represents the results at different time delays, but at a fixed bath temperature of 1,3K. The Cu 2O sample of [100] orientation was provided by A.Revcolevschi, University of Paris IV.

129.0

128.8

128.6

128.4

Energy, meV Energy, 128.2

0 10 20 30 40 50 Bath temperature, K

Fig. 2.33. 1s-2p paraexciton transition energy as a function of temperature.

Chemicalpotential - 1.0 a b 20

0.4 µ Temperature, Temperature, lnK , meV , 3 0 1000 2000 3000 0 1000 2000 3000 Time, ps Time, ps Fig. 2.34. Time dependence of (a) the exciton temperature and (b) chemical potential from the fitting analysis of the experimental results by Eq. 2.34.

-3

a b 10 n, Density, ry cm a 55 7 nd 17 u bo n tio si n tra e 20 as ph

3 cm 7 Density,n, 10 -3 0 1000 2000 3000 7 20 55 Time, ps Temperature, K

Fig. 4.35. (a) Time dependence of the exciton density, calculated using Eq. 1.18 And results from Fig. 2.34 (solid line is the guide for an eye). (b)n,T phase diagram (solid line is the phase transition boundary).

74 This is in a good agreement with the absorption and luminescence experiments [39] on orthoexciton states, assuming the splitting of the ortho and para states is temperature independent. The gas temperature resulting from fitting the convoluted Eq. 2.34 to time-resolved spectra (Fig. 2.32) of excitonic transitions are shown in Fig. 2.34a. With respect to the orthoexcitons dynamics the cooling of the gas occurs with a relatively long decay time of 7 ns. The decay of the chemical potential in time, shown in Fig. 2.34b, is also found to be slower than for the orthoexcitons (Fig. 2.3), resulting in a time constant of 9 ns. Similar to the orthoexciton case, we can calculate the density of the paraexciton gas using the Eq. 1.18 . The result is shown in Fig. 2.35. The state of the gas is approaching the BEC boundary, but the lifetime of the particles is still not sufficient to cross the boundary. Actually, Kavoulakis [10] has argued that the above mentioned fitting-based-analysis which was used to extract the exciton temperature and, therefore, density is misleading since the gas is in a classical regime. In general, the condition of which model, classical or quantum, should be considered as correct is based on the relation between the thermal de Broglie wavelength a of the particles and average distance between them l=n 1/3 , where n is the gas density [110]. We can establish the transition condition from classical to the quantum case as:

a << l for classical regime a >> l for quantum regime

The border equation will determine the “transition” concentration:

3  p   mk T  /3 2   B nQ ≡   = g 2  , (2.35)  h   2πh 

This, so called quantum density corresponds to one particle in a cube with sides equal to the average de Broglie wavelength. The estimated value (Fig. 2.35 a,b) is around the quantum density for a paraexciton gas with a particle density of 10 -18 cm -3 at 7K. This means that we can already consider our gas being in the quantum regime. The above obtained result can even be easily checked using a different method of density estimation. This analytical method is based on the direct connection of the induced absorption with oscillator strength. By calculating the Einstein coefficients, the dielectric function and relating these to the oscillator strength, it can be shown [111], that the oscillator strength is given by:

75 2m∆E 2 f = µ , (2.36) i h2e2 i where ∆E is the transition energy. From the dielectric function with simultaneous laser excitation we take the additional imaginary part ε*, since it is determines the induced absorption [38, 39]:

  ε 2* = Im []ε (E) - ε * (E) = Im ∑ χi , (2.37)  i  2 2 n1sh e fi χi (E) = 2 2 mε 0 ∆E − E − ihΓE

where Γ is the damping and n1s is the density of the 1s excited state. Now the induced absorption can be written as:

2E n µ 2 hΓ ε 2 (* E)E Im []χi (E) E i 1s i 4 αind ,i (E) = ≈ = , (2.38) 2 2 hcn hc ε B hcε 0 ε B ∆E − E + hΓ () ()2

Here c is the light velocity and the index of the refraction is approximated by εB=7,1, [39, 44]. The ratio on the right hand side describes a normalized Lorentzian with an integrated area of π/2 Integration of the Eq. 2.38, assuming that ħΓ<< ∆E, yields the connection between the density of the 1s excited state and the area of the induced absorption (Fig. 2.18, 2.21a):

hcε 0 ε B Ai n1s = 2 π µi ∆E

where Ai = ∫αind ,i (E)dE , (2.38)

76 Because the 1s exciton differs from the hydrogen model, the matrix dipole element needs to be calculated specially, taking into account the wave functions of the 1s and 2p excited states. The exact value of the dipole matrix element was taken from [38, 39].

1.2 -3 1.0 cm 18 0.8

0.6

0.4

Density, 10 0.2

0.0 0 1000 2000 3000 Time, ps Fig. 2.36. Time dependence of the exciton density, calculated using Eq. 2.38; open circles - 2,07 eV one-photon absorption excitation method, closed circles - 1,55 eV two-

photon absorption excitation method. Sample temperature T=8K. Cu 2O crystal of [100] orientation was provided by A.Revcolevschi (University of Paris IV).

As a result of applying the described method to the experimental data, the early time dynamics of the 1s-2p paraexciton transition are shown for two different excitation methods in Fig. 2.36: exciton gas created via two-photon absorption into the blue series and direct one-photon absorption to the 1s yellow orthoexciton level. It is obvious, that these results agree well with one discussed above and presented in Fig. 2.35a. Also, as discussed in Chapter 1, different methods of excitation may excite different excitonic states, with different efficiencies. Besides the difference in the exciton creation mechanism the penetration depth differs: 30 µm for 2,07 eV excitation and the whole bulk in a case of 1,55 eV. It is obvious, that one can reach much higher excitation densities via one-photon absorption, which very well agrees with the experimental results (Fig. 2.36). Moreover, we can make an additional conclusion from Fig. 2.36. Since the creation of the exciton gas via two-photon excitation with 1,55 eV pulses is going through the blue series, one may expect that the 1s paraexciton state can be created via decay to the yellow series. For one-photon absorption of 2,07 eV pulses the only possibility to populate the paraexciton state is the spin-flip down-conversion from the orthoexciton state. The present experimental results (Fig. 2.36) are showing that despite the excitation method, 1s paraexcitons are primarily created via down-conversion from the 1s orthoexciton state.

2.2.6. Paraexciton gas parameters evolution in ( µTn )-space.

Above we established the ways how to determine exciton gas statistical parameters. Now we will discuss question: “Can we reach the condensed state for paraexcitons?” In a case of non-condensed gas, the chemical potential, temperature and density are connected: ∞

n(µ,T) = A∫ dEf BE (E, µ,T)D(E) 0 where fBE is the Bose-Einstein distribution function, D(E) is the density of states. Therefore, the possible states of a boson gas form a surface in µTn-space (Fig. 2.37): knowing two parameters, one can easily predict the third. In the experiment, we initially create a hot exciton gas of high density. In time, the gas temperature will decay toward the lattice temperature, leading, in principle, to a decrease of the chemical potential. If the density of

Fig. 2.37. The dynamics of the paraexciton gas statistical parameters. Light grey trace – calculated dynamics (lifetime 1 µs); dark grey – dynamics obtained from the described above experiments.

78 the gas stays sufficiently high, the exciton gas will reach the condensate state: the surface

(n,T) µ=0 determines the Bose-Einstein condensation boundary. The dark grey trace in Fig. 2.37 shows the real dynamics of the exciton gas statistical parameters as determined from the experiments described in §2.2.4. At early times, the exciton temperature and chemical potential decay providing a “ movement ” toward the transition boundary. However, in these experiments the lifetime of the paraexcitons was relatively small (10 ns), leading to fast density decay and a chemical potential which “moves” toward larger negative values. As a result, the trace of exciton gas state is slowly turning away from the condensation boundary. For comparison, we did a theoretical calculation for a long-lived paraexciton gas (light grey trace in Fig. 2.37). The lifetime of the particles was assumed to be 1 µs, while the gas cooling time was chosen to be the same as obtained from the present experiment. The result of the calculation does show, that for such a gas condensation may occur (in this case after 1000 ns), if the initial exciton gas density is high enough.

2.3. Conclusions.

In conclusion, we investigated two types of excitons in cuprous oxide and discussed the ideas and results of the experiments aimed at determining the exciton gas parameters and their time evolution. It was shown that the lifetime is one of the most critical parameters determining whether a condensate state may occur. For the orthoexcitons, it was shown that the gas adjusts its quantum properties to the density and temperature and that it is in quasi-equilibrium. Apparently the particle decay is faster than the cooling rate, which inhibits a BEC transition. Further, the ways of direct and in-direct probing of the optically inactive paraexcitons were presented and described. Based on the results of the experiments and analysis of the experimental data, we discussed the processes which may contribute to paraexciton losses. The most important conclusion is that at the currently achieved densities (~10 18 cm -3) the main paraexciton loss process is trapping by crystal imperfections. It was concluded, that the lifetime is strongly depends on amount of copper impurities and a mechanism of exciton trapping by copper impurities was proposed. The presented experiments were carried out using quite impure samples: the cooling rate was not sufficient to overcome the particle loss. However, under the proper experimental conditions, it is possible for the paraexcitons to achieve a condensate state: low bath temperature (1,2K for densities ~10 17 cm -3), low concentration of crystal defects, one-photon excitation method for a smaller excitation volume and intense enough excitation source.

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83 Chapter 3

Exciton gas in a high magnetic field

In this chapter we present and discuss an experimental study of the yellow exciton series in Cu 2O in the presence of a strong magnetic field. Magneto-luminescence spectra of the exciton gas in fields up to 32 T show an activation of the direct and phonon-assisted paraexciton luminescence due to mixing with the quadruple allowed orthoexciton state. This method of activation of the optically inactive paraexciton does not dramatically reduce the lifetime, which is one of the critical parameters for the possibility of exciton Bose- Einstein condensation. Furthermore, the observed phonon-assisted luminescence yields direct information on the statistical distribution of occupied states, giving access to both the chemical potential and the exciton gas temperature. Magneto-absorption experiments show a complex behavior of the orthoexciton series in the presence of a magnetic field resulting from a combination of Zeeman splitting, diamagnetic shifts, and level mixing. For energies well above the band-gap, the magnetic energy is the determining factor leading to the appearance of Landau level like absorption peaks in the magneto-absorption spectra.

84 3.1. Magneto-luminescence of exciton gas.

Since the 1s paraexciton is optically inactive, one has to break the symmetry in order to make the decay of the paraexciton observable in optical experiments. These kinds of experiments are important since, for instance, the luminescence spectrum may yield direct information on the statistical distribution of occupied states. There are several methods to break the cubic symmetry in cuprous oxide, including the application of pressure [1-3] and electric fields [4]. Pressure experiments have the advantage of limiting the diffusion throughout the sample by the creation of a trap for excitons. The lifetime is however usually reduced by orders of magnitude [5] which is not favorable for condensation. An applied magnetic field breaks the symmetry in a more subtle way. In this case, the optically forbidden paraexciton state |S> mixes with the quadruple allowed orthoexciton m J=0 state

|T 0>. This leads to a weak, field tunable emission from the paraexciton state, proportional 2 2 to (ge − g h ) B . Some experiments on excitons in a magnetic field have already taken place [6-8]. These works were mainly devoted to the investigation of the magneto- oscillatory absorption, energy structure and dynamical band-gap [6, 9-11]. A detailed discussion of the magneto-absorption of Cu 2O may be found in §3.2, whereas the present section focuses on the magneto-luminescence of exciton gas. The first investigation of magneto-luminescence of paraexcitons was reported in [12]. The observation of the direct paraexciton emission in this experiment deemed problematic due to the chosen experimental geometry. In particular, luminescence from phonon-assisted transitions was to small to make any quantative statistical conclusions. Apart from using the optimal experimental geometry, it is also favorable to perform the experiments in high magnetic fields since the intensity of the paraexciton luminescence scales quadratically with the applied field.

3.1.1. Experimental details.

Circularly polarized magneto-luminescence experiments have been performed in a reflection Faraday geometry in magnetic fields up to 32 T generated by the M9 resistive magnet at the Grenoble High Magnetic Field Laboratory, with the sample placed in a bath cryostat (T = 1.2 K). A solid state laser (532 nm) was used as an excitation source. The emitted light was detected with a double monochromator (resolution 0.05 nm) equipped with a LN 2 cooled CCD detector. Since the luminescence efficiency and the degree of polarization depends strongly on the relative orientation of the crystallographic axes with the light wave vector [13, 14, 15],

85 we studied cuprous oxide single crystals of [110] and [100] orientation, grown by a floating zone technique [16], cut and polished into platelet of 500 µm thickness. For the static luminescence and absorption experiments the sample was provided by A.Revcolevschi (University of Paris IV). For the time-resolved magneto-luminescence the high-purity sample was used (the sample was kindly provided by M.Gonokami, University of Tokyo).

3.1.2. Experimental results.

Luminescence spectra recorded at T=1.2K are shown in Fig. 3.1, for both B=0 T (unpolarized), and for B=32 T ( σ+ and σ- polarized). As expected, the exciton luminescence spectrum exhibits only orthoexciton features at B=0T (Fig. 5.1a).

a) B = 0 T O O 2 0 O 3 O 1

+ b) B = 32 T, σσσ direct para

m =-1 P m =1 P P O J 0 O J 3 2 P 2 m =-1 0 1 O J Intensity 0

c) m =1 - direct para O J B = 32 T, σσσ 0 P 0 m =-1 m =1 J P J O 3 P P O 0 2 1 2

2.01 2.02 2.03 Wavelength, nm

Fig. 3.1. Luminescence spectra of excitons in Cu 2O for [110] orientation at T=1,2K a) − Luminescence in absence of a magnetic field: O 0 – direct orthoexciton emission, O 1 – Γ5 − (6,6 meV) phonon-assisted orthoexciton emission, O 2 – Γ3 (13,2 meV) phonon-assisted − orthoexciton emission, O 3 – Γ4 (18,2 meV) phonon-assisted orthoexciton emission; b)

Luminescence in a magnetic field strength B=32 T, right circular polarization: P 0 direct − − paraexciton emission, P 1 ( Γ5 , 6,6 meV) phonon-assisted paraexciton emission, P 2 ( Γ3 , − 13,2 meV) phonon-assisted paraexciton emission, P 3 ( Γ4 , 18,2 meV) phonon-assisted paraexciton emission; c) Magnetic field strength B=32 T, left circular polarization.

86 + Apart from the quadruple allowed Γ5 direct exciton transition (O 0 at 2.033 eV), there are clear lines from phonon-assisted orthoexciton transitions with the simultaneous - - - excitation of Γ5 (O 1 at 6.6 meV), Γ3 (O 2 at 13.2 meV), and Γ4 (O 3 at 18.2 meV) phonons, [17]. The threefold degeneracy of the triplet orthoexciton state is lifted in a magnetic field, as is clearly observed in the σ+ and σ- polarized spectra in Fig. 3.1b and c. It is important to note that due to the experimental conditions the spectra are only 85 % polarized as is + - evidenced by the presence of the m J = -1 line in the σ -spectra and the m J = 1 line in the σ - spectra. Both the direct and the phonon-assisted orthoexciton emissions split into their m J + - components, and the σ and σ spectra show predominantly the m J=1 and m J=−1 components, respectively. These magneto-luminescence spectra also show the apparition of magnetic field activated paraexciton recombination, which appears as a sharp feature at

2.0216 eV (P 0). Of more importance for possible studies of Bose-Einstein condensation of excitons in Cu 2O, we also observe three phonon-assisted paraexciton recombination lines - - - (P 1, P 2, P 3) at 2.011 eV ( Γ5 phonon), 2.083 eV ( Γ3 ) and 2.045 eV ( Γ4 ), respectively. As the activation of the paraexciton luminescence occurs via the coupling to the m J=0 orthoexciton level, the paraexciton emission is not circularly polarized. This is demonstrated in more detail in Fig. 3.3a, which shows equal intensities for the phonon assisted para-exciton emission in both polarizations. A two dimensional color map of the magnetic field dependence of the 1s direct and phonon-assisted orthoexciton emission is shown in Fig. 3.2(a,b). At B=0T, a single line is + observed corresponding to the emission of the degenerate triplet (Γ5 ) exciton. For + - increasing magnetic field, this line shows a splitting into its m J=1 ( σ ) and m J=-1 ( σ ) components and a weak emission from the m J=0, due to the experimental conditions (Fig. 2 2 a). The m J=0 level shows a clear diamagnetic (~B ) shift. This effect is discussed below in more details. As Fig. 3.2b shows, the splitting also occurs for the phonon-assisted - orthoexciton transitions. It is mostly evident in the Γ3 phonon-assisted line. Above B=6 T, the direct paraexciton emission becomes observable in the luminescence spectrum as a result of a perturbation which can be written as: 1 V = µ (g S z B + g S z B ), (3.1) 2 B e e z h h z

Fig. 3.2. (a) 2D map of spectra field dependence in 1s orthoexciton direct recombination energy range at magnetic fields up to 32T (1T step) and T=1,2K (b) – 2D map of spectra field dependence in 1s orthoexciton phonon-assisted emission energy range at magnetic fields up to 32 T and T=1,2K. Crystallographic orientation [110].

88 The presence of this term results in a change of both the paraexciton |S> and orthoexciton |T 0> wave functions as well as their energies as:

0 (ge − gh )µB B 0 S ⇒ S − T0 2 2 2 2 ∆ + ∆ + ()ge − gh µB B

2 2 2 2 0 ∆ − ∆ + ()ge − gh µB B 0 T0 ⇒ T0 − S , (3.2) ()ge − gh µB B

∆ + ∆2 + ()g − g 2 µ 2B2 E ⇒ E0 − e h B s S 2 2 2 2 2 0 ∆ − ∆ + ()ge − gh µB B ET ⇒ E + , (3.3) 0 T0 2

where ∆=12 meV is the energy difference between the ortho and para levels at B=0 T, ge 0 0 and gh are the electron and hole g-factors, |S> and |T 0> is the unperturbed paraexciton and mJ=0 orthoexciton wave functions, respectively. The transition probability P of the radiative decay changes as:

2 2 P = 0 ⇒ P ~ (ge − gh ) B , (3.4)

Indeed, the intensity of the paraexciton emission increases as B2. However, it is not possible to extract any thermodynamic information from the direct paraexciton emission.. More interesting is the observation of the three distinct phonon-assisted paraexciton - - emissions P 1, P 2, P 3 at energies respectively of 2.011 eV ( Γ5 phonon), 2.083 eV ( Γ3 ) and - 2.045 eV ( Γ4 ) as shown in Fig. 3.3a The paraexciton direct emission as well as the three phonon-assisted replicas are linearly polarized. The quadratic behavior of the paraexciton emission intensity is indeed observed experimentally, as shown in Fig. 3.3b which displays - the intensity of both the direct emission as well as the Γ3 phonon-assisted paraexciton line as a function of B2. The energy shifts predicted by Eq. 3.3 and Eq. 3.4 for the level are also observed, as shown in Fig. 3.3c. However, the orthoexciton shows a much weaker field dependence, presumably due to the interaction with a higher lying level, [13].

The observed splitting yields a g-factor of | ge+gh| = 1,66, where ge, gh are the g-factors for the electrons and the holes, respectively. This agrees well with the value obtained by

89 Certier et al. g=1.65 [18], Kuwabara et al. g=1.66 [19], Frohlich et al. g=1.64 [20] and Kono et. al. g=1.63 [15] within the experimental error. However it is slightly different from the value reported by Langer et al. , g=1.78 [21].

2 Energy, eV Magnetic field, T 2.005 2.010 0 100 200 300 400 500 600 700

(a) (b) Intensity, arb. units P3 P2

P 1

Intensity, arb. units arb. Intensity, 2.0355 2.038

direct orthoexciton (c) direct orthoexciton (d) 2.036 2.0350 Energy, eV 2.034 ∆=0.31 meV

2.0345 2.032

2.024 direct paraexciton direct paraexciton

Energy,eV 2.0225 2.022 ∆=0.57 meV 2.020 2.0220 0 100 200 300 400 500 0 10 20 30

Magnetic field, T Magnetic field, T 2

Fig. 3.3. (a) Positions of exciton lines as a function of magnetic field: closed circles – orthoexcitons, open circles - paraexcitons. Crystal orientation [110]. (b) Paraexciton direct

emission and m J=0 orthoexciton line energy position versus the squared magnetic field strength, obtained from experiments on a [100] oriented platelet at T=1.2K. The solid line is a linear fit to the data. (c) P1 ( Γ5), P2 ( Γ3), P3 ( Γ4) phonon-assisted paraexciton emission at 32 T and T=1,2K: open circles – right circular polarization, closed circles – left circular - polarization. (d) Intensity of the Γ3 phonon-assisted paraexciton emission line (closed circles) and direct paraexciton emission line (open circles) versus the squared magnetic field strength, obtained from experiments on a [110] oriented platelet at T=1.2K. The solid line is a linear fit to the data. the coincidence in energy of the direct paraexciton emission with orthoexciton phonon-assisted replicas makes the observation and analysis in low magnetic fields more complicated.

90 3.1.3. Paraexciton kinetic energy distribution.

In order to find the information characterizing the paraexciton kinetic energy distribution, we now analyze the phonon-assisted emission line, [22, 23]. The lineshape of this emission can analytically be described by (see also Chapter 1):

(E −E ) 2 ∞ − 0 Γ 2 I(E0 ) = A ∫ D(E) f (E)e dE , (3.5) −∞ where D(E) is the density of excitonic states, Γ is the spectral resolution of the experimental setup and f(E) is either the Maxwell-Boltzmann or Bose-Einstein distribution function. The Bose-Einstein distribution function is applicable when the paraexciton density is high or when the temperature is relatively low. We have fitted Eq. 3.5 to the data in Fig. 3.4 using both the classical (grey line) and the quantum (black line) distribution. As the Fig. 3.4 shows, both distributions fit reasonably well to the data, even though the Bose-Einstein distribution seems to give a slightly better fit. From these fits, we can deduce an excitonic temperature of T BE =8K and a chemical potential value of µ=-0.5 meV using the Bose-

Einstein distribution, or a temperature T MB =7.3K using the Maxwell-Boltzman distribution.

para Γ− 3

para Γ− 5 Intensity, arb. units Intensity, arb.

2.008 2.010 2.012 Energy, eV - Fig. 3.4. Γ3 phonon-assisted paraexciton emission at bath temperature T=1.2K and

magnetic field B=23T (Cu 2O sample of [110] orientation). Open circles - experimental data; black line - theoretical model using Bose-Einstein distribution (T=7K, µ=-0.5meV ); grey line - theoretical model using Maxwell-Boltzman distribution (T=6,5K).

91 In general, if the particles obey a Bose-Einstein distribution, the number of particles per unit volume will be [22, 23]:

3 g  2m  2 ∞ E /1 2 n =   dE , (3.7) 2 2 ∫ (E −µ /) k BT 4π  h  0 e −1

From the obtained values of the chemical potential and excitonic temperature, the density of paraexcitons calculated using Eq. 3.7 is of 5×10 −17 cm −3. The experimental conditions for which a quantum or a classical model has to be considered are determined is discussed in §1.2. In the present experiment, this estimated value is 3 times smaller than the quantum density (~2×10 −18 cm −3). This means that the paraexciton gas can still be considered as being in the classical regime, but very close to the transition limit.

3.1.4. Time-resolved magneto-luminescence experiments.

The results of the experiments discussed above provide a unique opportunity to directly determine the thermodynamical properties of the paraexciton gas, which could be used as indication of Bose-Einstein condensation of the paraexciton gas in time-resolved experiments. In time-resolved experiments we used almost the same experimental conditions, as described in section 3.1.1. Magneto-luminescence experiments have been performed in a reflection Faraday geometry in magnetic fields up to 32 T, with the sample placed in a bath cryostat (T = 1.2 K). A low impurity sample was cut and polished into a platelet of 500 µm thickness oriented in [100] crystallographic direction. A pulsed solid state laser (40 ns, 4 kHz repetition rate, 523 nm) was used as an excitation source. The emitted light was detected with a single monochromator (resolution 0.1 nm) equipped with a fast CCD detector. The time resolution was provided electronically, by switching the CCD detector on/off at certain time, after the excitation pulse. Fig. 3.5 shows time-resolved magneto-luminescence spectra of yellow excitons at T=1,2

K for B=28 T on Cu 2O sample in [110] crystallographic direction. At early times the spectrum shows the presence of both ortho and paraexcitons. In time the orthoexcitonic features vanish, due to the short particle lifetime (see Chapter 2). We established, that the lifetime of the para excitons is long in high fields (up to 1 µs). The fast initial decay of the spectrum intensity (~60 ns) is due to the pump pulse time-duration (40 ns), meaning, that the spectrum intensity simply follows the pump pulse time profile (Fig. 3.6a).

Direct paraexciton Phonon-assisted emission paraexciton transitions

Γ − 3 orthoexciton Intensity Γ − 3 Γ − 5

T 0 im ,5 e , µ s 1 2.005 2.010 2.015 2.020 Energy, eV

Fig. 3.5. Exciton luminescence at different times after the excitation with 40 ns pulse of 523

nm wavelength. Bath temperature T=1,2 K and field strength B=28T. Cu2O sample of [110] orientation was provided by M. Gonokami (University of Tokyo).

Time, ns Energy, eV 0 500 1000 2.01 2.02 2.03

a paraexciton b direct emission paraexcitons Intensity orthoexcitons at time t=600 ns

Intensity τ=510 ns paraexciton phonon-assisted emission

1.0 Chemical potential - 20 c d τ=350 ns τ=430 ns 7

0.4

3 µ, meV Temperature, K Temperature,

1 0.1 200 400 600 200 400 600 Time, ns Time, ns

Fig. 3.6. (a) Time evolution of the intensity of paraexciton (circles) and orthoexciton phonon-assited lines. (b) The magneto-luminescence spectrum of the paraexciton gas at 600 ns after the excitation. (c,d) Evolution of the thermodynamical paraexciton gas parameters with time: temperature (c) and chemical potential (d). Excitation with 40 ns pulse of 523 nm wavelength. Bath temperature T=1,2K. . Cu2O sample of [110] orientation was provided by M. Gonokami (University of Tokyo).

We can extract the values of the gas statistical parameters for each time point after the excitation pulse, from the experimental data, using the method described in §3.1.3. The evolution of the paraexciton gas temperature and chemical potential in time is presented in Fig. 3.6(c,d). Both of these parameters reveal a one-exponential decay. Unfortunately, the analysis of the data at times around 500 ns after the excitation pulse and longer is difficult, due to the small intensity. The width of the phonon-assisted line at 500 ns after excitation pulse is mainly limited by the resolution of the detection system being 0,1 nm. (Fig. 3.6b). However, the preliminary simple estimation of gas density at 600 ns (after gas thermalization) is around 10 16 cm -3. For such densities, the transition temperature is around 0,5K which is below the experimental bath temperature.

94 3.2. Magneto-absorption of the excitons.

As discussed in Chapter 1, the absorption spectrum of cubic Cu 2O exhibits a discrete structure comprising a number of exciton series of which the hydrogen-like series located in the yellow region of the spectrum is most studied (the Wannier-Mott yellow exciton series),

[24]. There is a long history of magneto-absorption studies carried out on Cu 2O. Experimentally, Gross and Zakharchenya have first observed the magnetic field dependence of the yellow excitons at low magnetic fields, [25]. Sasaki and Kuwabara carried out more precise measurements of the magnetic field dependence in a wide range of magnetic fields up to 16T, [11]. They found that the magneto-absorption spectrum has a rather complicated structure. For low fields the absorption peaks in the low energy range can be well explained by the hydrogen model, taking into account the central cell correction for the 1s state. With increasing energy or magnetic field, interaction of these hydrogen-like states occurs and mixing of excitonic levels takes place, [11]. Above the ionization threshold of the excitons, many equally spaced absorption peaks were observed, [11, 25]. Sasaki’s experiments have been extended to higher magnetic fields (up to 25 T) and better spectral resolution in experiments by Seyama et. al. , [26]. At very large magnetic fields (up to 150 T), the cyclotron mass and the Zeeman splitting of Cu 2O has been determined by Kobayashi et. al. in their observation of n=2 and n=3 exciton peaks in pulsed magnetic fields. In this work the transition from a quadratic diamagnetic shift to a linear magnetic shift of Landau levels was observed, [27]. This observation clarifies the basic properties of absorption peaks in applied magnetic fields, however, all the measurements were performed at rather large intervals of magnetic field and detailed magnetic field dependence is missing. Indeed, to clarify the developments of the absorption levels due to applied magnetic field, precise measurements of position and intensity of each level with continuous change of magnetic field is needed. In this part, we will present the results of detailed measurements of magneto-absorption spectra of Cu 2O for an extended range of static magnetic fields up to 32T. The reader can find the discussion of the experimental results being quite descriptive: further data analysis is yet to be done.

3.2.1. Experimental details.

We studied cuprous oxide single crystals of [100] orientation, grown by a floating zone technique. The sample was kindly provided by A. Revcolevschi (University of Paris IV). Polarized magneto-absorption experiments were performed on cut and polished platelets of 30 µm thickness. The experiments were performed in a Faraday geometry in

95 magnetic fields up to 32 T generated by the M9 resistive magnet at the Grenoble High Magnetic Field Laboratory, with the sample placed in a bath cryostat (T = 1.2 K). A halogen lamp was used as a light source. The transmitted light was detected with a double monochromator equipped with a LN 2 cooled CCD detector (resolution 0.02 nm).

3.2.2. Results and discussion.

The description of the Cu 2O absorption spectrum in an absence of the magnetic field is presented in Chapter 1. Here we will focus only on its magneto-induced features. Fig. 3.7 exhibits the magneto-absoprtion spectra of Cu 2O at B=0, 10 and 20 T. Left panels represent left circular polarized part of the spectra and right panels represent right circular polarized spectra.

2.16 2.19 2.16 2.19

− 0T, σ+ 0T, σ

n=3 n=3 n=2 n=2 Optical densityOptical

+ 10T, σ 10T, σ−

+ Optical Optical density 20T, σ 20T, σ−

continuum continuum

2.16 2.19 2.16 2.19

Energy, eV

Fig. 3.7. Magneto-absorption spectra of Cu 2O in yellow exciton energy range for B=0, 10 and 20T. Left panels: left circular polarized spectra; right panels: right circular polarized spectra. Spectra are only 85% polarized, due to the experimental conditions. T=1,2K, [100] crystal orientation.

Again, as already noted in section 3.1, the spectra are only 85 % polarized, as + - evidenced by the presence of the m J = -1 line in the σ -spectra and the m J = 1 line in the σ - spectra. In absence of a magnetic field, the well-known hydrogen-like spectra is observed. We can observe at least 4 exciton peaks of the yellow series (n=2-5). With application of the magnetic field, the hydrogen-like spectra show a splitting with the field. The picture becomes more complex at high magnetic fields. A complicated magneto-oscillatory absorption above the threshold energy of the inter-band transition at high magnetic fields is observed (Fig. 3.7). In order to address the complexity of the spectra, we performed an experiment measuring the detailed field dependence of the circularly polarized spectra up to B=32 T. Fig. 3.8 represents an overview of the optical absorption spectra in a colour image plot of the intensity as a function of the photon energy and the magnetic field. In this plot, red parts represent the higher absorbance regions. The left part represents the σ--spectra, whereas the right part represents σ+-spectra. For the n=2 state, the position of the absorption peak shifts simply with magnetic field for the whole magnetic field range (Fig. 3.8). Above 10T, this peak splits into two peaks of different values of the magnetic quantum number m. The behavior of the absorption peaks gradually complicates with increasing of the main quantum number n. The general behavior of the absorption peaks can be described as follows, [11]. In case of Cu 2O, only p-states transition for each n-index are dipole-allowed. We already observed these states in the absence of a magnetic field. With increasing magnetic field, one can observe other l- index (orbital number) states, due to the finite off-diagonal elements induced by the magnetic field, [26]. Finally, the energy region in the vicinity and above the band-gap energy is particularly interesting (Fig. 3.8). Already at 8T equidistant quasi-Landau levels become visible above the band gap. For higher n excitons the excitonic radius becomes comparable to or even larger than the magnetic length. A description in terms of perturbed excitons is then no longer applicable. Hammura et al. [28] suggested, that the electron-hole pair undergoes a periodic orbit mainly determined by the Coulomb potential which is perturbed by the presence of a magnetic potential.

98 The main features of the experimentally observed magneto-optical absorption pattern can be explained by considering the Hamiltonian of the interacting electron-hole pair in a magnetic field. Let’s first write the Langrangian L of two oppositely charged particles: m r& 2 m r& 2 e e L = e e + h h − A(r ) + A(r ), (3.8) 2 2 c e c h where we used the symmetric gauge for magnetic field H in z-direction: 1 A(r) = []H × r , (3.9) 2

For future discussion, it is useful to change coordinates into coordinates for the center of mass and for the relative motion:

 r m + r m R = e e h h  me + mh  r = re − rh  m r = R + h r  e M  , (3.10) m r = R − e r,  h M

where M=m e+m h. For these new coordinates, we can re-write the Langrangian L:

99 L = LR + Lr + LrR MR& 2 L = R 2 2 2 µr& 1 mh − me e Lr = + A(r)r& + 2 c mh + me εr e L = ()[][]H × R r& + H × r R& , (3.11) rR 2c

where µ is the exciton reduced mass and ε is the dielectric constant of the medium. The adding the Eq. 5.12 into the Eq. 5.11 does not change the motion law, [29], thus we can eliminate R-dependent term here:

d ()[]H × R r = []H × R r& + [H × R&]r, (3.12) dt

e d e L'= L − ()[]H × R r = ([]H × r R& ), (3.13) rR 2c dt c

Finally, the Langrangian L’ can be expressed as follow:

MR& 2 e µr&2 e m − m e2 L'= + ()[]H × r R& + + []H × r r& h e + , (3.14) 2 c 2 2c mh + me εr

Now the Langrangian does not depend on R. Its conjugated momentum can be written as:

100 ∂L' e P = = MR& + []H × r , ∂R& c ∂L' e p = = µr& + []H × r γ (3.15) ∂r& 2c

where γ=(m h-me)/ (m h+m e). To quantitize the problem, we write the Hamiltonian:

MR& 2 µr&2 e2 H = PR& + rp& − L'= + − = 2 2 εr 1  e 2 1  e 2 e2  P − []H × r  +  p − []H × r γ  − , (3.16) 2M  c  2µ  c  εr

Eq. 3.16 represents the Hamiltonian of a particle of mass µ, moving in the effective magnetic field γH and in the potential field, [29]:

1  e 2 e2 U (r) =  P − []H × r  − , (3.17) 2M  c  εr

In a case of optically created excitons, the total exciton momentum equals the photon momentum, which is negligible. Eq. 3.17 can then be further simplified to:

1  e 2 e2 U (r) =  []H × r  − , (3.18) P=0 2M  c  εr

For the simplicity, we choose the z-axis along the magnetic field H and split r into components parallel and perpendicular to the magnetic field:

101 2 r r 2 2 2 2 ([H × r]) = ((− ri y + rj x )H ) = (rx + ry )H r r r r = rk z + ⊥ ,r (3.19)

This Hamiltonian is invariant with respect to rotations around the z-axis. The corresponding projection of the momentum (projection to the direction of H) is conserved. We, finally, can write:

e2 H 2 e2 U (r) = r 2 − , (3.20) P=0 2Mc 2 ⊥ εr 1  e 2 e2 H 2 e2 H =  p + []γH × r  + r 2 − = 2µ  2c  2Mc 2 ⊥ εr p2 e e2γ 2 H 2 e2H 2 e2 + γH ⋅[]r × p + r 2 + r 2 − , (3.21) 2µ 2µc 8µc2 ⊥ 2Mc 2 ⊥ εr

Using common notation ħl=[r×p] , we can rewrite Eq. 3.21 as follows:

p2 r r µω 2r 2 e2 H = + µ *γH ⋅l + ⊥ − , (3.22) 2µ B 8 εr Here eh m eH 4µ µ * = = µ , ω = γ 2 + B 2µc B µ µc M  1 1  p2 = −h2 ∂ 2 + ∂ + ∂ 2 + ∂ 2 ,  r⊥r⊥ r⊥ 2 ΘΘ zz   r⊥ r⊥ 

Making the substitution ψ (r⊥ ,θ, z) = f (r⊥ ,θ, z /) r⊥ , we get rid of the /1 r (∂ ) term: ⊥ r⊥

102  1  f (r ,θ , z) 1  1  ∂ 2 + ∂  ⊥ = ∂ 2 +  f  r⊥ r⊥ r⊥   r⊥r⊥ 2   r⊥  r⊥ r⊥  4r⊥ 

Since lz commutes with Hamiltonian, we can find the eigenfunctions in the form im θ ψ(r ┴,θ,z)= e φ(r ┴,z) , leading to:

 h2 h2  1  µω 2r 2 e2  - ()∂2 + ∂2 + m2 −  + µ *γH m + ⊥ −  f (r , z) =  2µ r⊥ r⊥ zz 2µr 2  4  B z 8 2 2  ⊥  ⊥ ε r⊥ + z 

= Ef (r⊥ , z), (3.23)

The potential energy is introduced in Eq. 3.24. For the sake of simplicity we introduce

the excitonic Bohr radius aB, the magnetic length λ, the Rydberg constant Ry , and the dimensionless coordinate r:

2 4 εh µe r⊥ hc aB = 2 , Ry = 2 2 , r = , λ = µe 2h ε aB eH

a 64444 74444 8 64444 7b 4444 8 2 1 2 4 U (r) m − 2 a a  4µ  = 4 − + γm B + B γ 2 + r 2 , 3.24 Ry r 2 2 λ2 4λ4  M  2  z  r +    aB 

Potential energy is constructed of two terms: (a) Coulomb interaction and (b) harmonic oscillator potential. For both of these problems separately, the semi-classical approximation is known to give the exact energy spectrum. The lowest eigenfunctions

(small n) are localized on a length scale aB, and the (b) term is not relevant for them: the lowest eigenvalues of the Hamiltonian are just eigenvalues of a hydrogen-like atom. When the Coulomb terms are dominating, we can treat the field-dependent terms pertrubatively. By averaging the diamagnetic term over the hydrogen atom wavefunction, we obtain a E~n 4 dependence for the peak energy shift. The nature of the large diamagnetic shift arises from rather large exciton Bohr radius. Such a result was also obtained before in [30]. Term

103 linear in H, will produce a linear Zeeman splitting thus lifting the degeneracy in lz The term non-linear with H can be taken into account in second-order perturbation theory. This term will produce a quadratic Paschen-Back effect. For higher energies the wavefunction “jumps out” from the Coulomb potential well and the characteristic size of the wave function becomes of the order of the magnetic length. Even higher in energy the eigenvalues are determined almost solely by the oscillator potential and spectrum becomes almost equidistant. In [31] it was shown that it is enough to consider the lowest Landau level to get a correct answer in first order of the small parameter ECoulomb /ħωc, where ECoulomb is the cyclotron energy. The wave function can then be written as:

Ψ(r) = φ(r⊥ )ψ (z) 1 − r 2  φ(r ) = exp  ⊥ , (3.25) ⊥ 4λ2 2πr0  

The function ψ(z) should satisfy the Schrödinger equation:

 2 2   h d   − 2 + U (z)ψ (z) = Wψ (z ,) (3.26)  2µ dz 

where U(z) is Coulomb energy, averaged over the motion in z-direction. To summarize, for a small energies when ECoulomb >ħωc we can use discussed above semi- classical approximation for the hydrogen-like atom model. In strong magnetic field when

ECoulomb <ħωc the spectrum should be formed by Landau levels. Selection rules allow only optical transitions with n →n, meaning that the exciton will be formed by hole and electron at the same Landau levels. We have solved the energy spectrum of Eq. 4.24 as a unction of magnetic field. Fig. 3.9 shows the experimental result already shown in Fig. 3.8 together with the calculated positions (solid lines). With increasing of magnetic field, two intense lines, originating from n=2, are clearly observed. These lines have different g-factors of 1.43 and -0.92. This fact doesn't allow us to attribute them to normal Zeeman splitting into m J=±1 components.

However, according to the selection rules two lines can be observed: m r=1, m J=0 and m r=0, mJ=1. One of the g-factors should be attributed to relative motion g-factor, g= γm/ µ, and another - to atomic g=g e+g h. These line scan be fitted with parabolas in accordance with expected linear Zeeman effect and quadratic Langevin magnetic field dependence (blue lines in Fig. 3.9).

104

Fig. 3.9. Image plot of the absorption spectra of Cu 2O at 1,2 K for [100] crystal orientation with preliminary theoretical fit. Left part – σ--polarization; right part – σ+-polarization. The red parts show higher absorption areas.

105 The same scenario we can apply for the level n=3, where these g-factors should be the same. The quadratic coefficients can be calculated in a straightforward way from the average over hydrogen wavefunction || 2. The interesting feature for n=3 level is the appearance of additional line (green line in Fig. 3.9). The fitting of this line gives a g-factor of -0,92, the same as for one from n=2 level, however, its quadratic coefficient is closer to one from n=3 level with g-factor of 1.43. This line was explained by Zhilich, Halpern and Zakharchenya in [7, 32], where they - - originated it with admixture of optically active Γ4 state to non-degenerate Γ2 level due to the presence of a magnetic field. This also explains the increase of the intensity of the line with the increase of the magnetic field strength. Further calculations have shown that the lines of n=4 level can be fitted by using the same parameters from discussed above three lines of n=2 and 3. Of course, at rather high magnetic fields, our curves go much higher in energy, because they represent only quadratic approximation. One, in fact, should use more complicated means than just perturbation theory to achieve better agreement. Basically, the terms we treat as perturbation become comparable with level distances, so perturbation theory is no longer valid. One more interesting thing attracted our attention is the appearance of the additional line at n=4 energy range. We attribute it to 4f ( mr=1) level, mixed with optically active 4p 2 2 (mr=1) state by Langevin term ~r ┴ H in our Hamiltonian (Eq. 3.22). This term has non- zero matrix elements from |nlm> to |n',l,m> and |n',l±2,m> and thus cannot produce additional visible levels in n=3, where we don't have f state. Here we discussed the experimental data and theoretical modeling presented in Fig. 3.8 and Fig. 3.9 rather qualitatively and preliminary. The result of the theoretical calculation does not yet fit well the experimental spectra and further data analysis is certainly required.

106 3.3. Conclusions.

We presented a study of the optical properties of the yellow exciton series in cuprous oxide in magnetic fields up to 32 T. We observed the luminescence of optical inactive paraexcitons due to mixing of 1s paraexciton state with 1s m J=0 orthoexciton state. The experimental results can be well explained using first order perturbation theory. We have demonstrated that an applied magnetic field can be used to reveal the paraexciton emission due to a gentle breaking of the symmetry. For this particular case, the lifetime of excitons is not strongly affected allowing efficient thermalization. Besides the direct emission from the paraexciton state, we also observed three additional lines which we identified as a result of the phonon-assisted emission. These lines provide a unique opportunity to directly determine the thermodynamical properties of the paraexciton gas which could be used as indication of Bose-Einstein condensation of the paraexciton gas. From our calculations we determined that the obtained exciton gas density is close to the critical gas density for which the gas is in the quantum regime. The results of the magneto-absorption study suggest that the hydrogen model can be applied only for states with small value of n. Deviations from the simple perturbation theory occur at the higher energies, but already at small magnetic field. For energies higher than the band-gap quasi-Landau levels are observed. This behavior of co-existence of Rydberg and Landau series might be understood in terms of the hydrogen model in high magnetic fields, which for hydrogen would mean fields as high as found in a white dwarf stars.

107 References

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109 Chapter 4

Induced terahertz response in Cu 2O

For several decades physicists have studied the dynamics of carriers in semiconductors. A very interesting region for these studies is the far-infrared, terahertz, or submillimeter region, where the frequency of the electromagnetic radiation is comparable to the carrier damping rate.

In this chapter we discuss the induced response of Cu 2O in the terahertz frequency region (1 THz ~ 4 meV). These results are also presented in [1]. Besides the experimental data and its preliminary discussion, we will depict the fundamental physics involved in generation and detection of terahertz pulse. The principles of terahertz time-domain spectroscopy are also discussed.

110 4.1. Terahertz time-domain spectroscopy.

A potential disadvantage of regular optical pump-probe experiments is that the probe energy, typically 1.5 eV or higher, is much larger than the relevant energy scales in many systems, such as the excitonic molecule energy scale in semiconductors. Combining THz time-domain spectroscopy with optical excitation one can measure the evolution of optically induced changes in the real and imaginary part of the dielectric function with a probe energy much closer to the relevant energy scale in the material. A typical example of mid-infrared reflectivity spectra calculated by M. Nagai et. al. [2] is displayed in Fig. 4.1. The reflectivity has a very distinctive response in the case of different concentrations and spatial distributions of the photo-excited carriers: it is flat for a low density of carriers (situation (1)); it has a Drude like shape for a high carrier density uniformly distributed over the sample (situation (2)) and it displays a resonance for a high density of carriers, gathered in ”droplets” inhomogeneously distributed throughout the sample. This example demonstrates the possibility of using optical-pump THz-probe spectroscopy in order to induce and see the signature of a metal to insulator transitions for example: in the case of a metal one would see a Drude like sample response, while in the case of an insulator a flat response will be observed.

Fig. 4.1. Typical response from different spatial distributions of photo-excited carriers in

an mid infrared pump-probe experiment. Reflectivity around plasma frequency ħωp at different distributions of electron-hole: (1) dilute excitonic gas, (2) homogenous plasma, and (3) electron-hole droplet, [2], (published with permission).

111 4.1.1. Generation and detection of picosecond terahertz pulses.

Since the emergence of Terahertz Spectroscopy, in the middle of the 80’s, there has been a continuous search for new materials with efficient emission of terahertz radiation. In parallel, an improvement process regarding the useful THz radiation bandwidth has taken place. As a result, today, a multitude of methods and materials are used for the generation of THz frequencies. Between them, the use of photoconductive antenna’s based on low- temperature-grown GaAs and Si-GaAs is the oldest method for generating and detecting THz pulses, [3-5]. THz radiation can result also from transient conductivity originating from high-intensity ultrashort laser pulses exciting the surface of an unbiased semiconductor [6]. Synchrotrons and free electron lasers can generate short pulses of far- infrared radiation, typically in the order of 5 -10 ps. It has been shown that laser-generated plasmas can produce radiation up to 4 THz, [7, 8]. More recently, there have been notable developments in using a single structure for transmitter and receiver, named transceiver and based on electrooptic crystals, [9]. In this case, the electrooptic terahertz transceiver alternately transmits pulsed electromagnetic radiation (optical rectification) and receives the return signal (electrooptic effect) using the same crystal. Still, probably the most popular choice for THz pulse generation, when working with amplified lasers, is to employ optical rectification in a nonlinear medium. The nonlinear medium is in most of the cases ZnTe but GaAs and GaP have been used as well. It has been shown that organic molecular crystals like DAST [10] and MBANP [11] are also capable of generating THz pulses by optical rectification. They are more efficient than ZnTe crystals of the same thickness, but have not yet found widespread use. In our spectrometer we create THz pulses by optical rectification in a 1 mm thick [110] oriented ZnTe crystal. The generation process is based on an amplified Ti:sapphire laser which provides pulses at 1 KHz repetition rate, with 150 fs pulse duration, and 800 nm wavelength. ZnTe is one of the most popular THz emitters used with near-infrared laser sources since being a wide bandgap semiconductor (Eg(ZnTe) = 2,28 eV) it is basically transparent at the laser pump wavelength, but shows good nonlinear properties. At 800 nm it has both large 2 nd order (2) −7 nonlinear susceptibility χ = 1,6 ·10 esu [12] and electrooptic coefficient r 41 = 4.04 pm/V [13]. The optical rectification process can be understood as the difference frequency analogue of second harmonic generation. In other words, when light interacts with a nonlinear medium and wave mixing between two frequencies, ω1 and ω2 occurs, the result is sum-frequency generation, ω1+ω2, and difference frequency generation, ω1-ω2. In the particular case when ω1=ω2, one generates both second harmonic and dc pulses. Because

112 the near-IR pulse has a duration of about 150 fs, a ”dc” pulse corresponding to the envelope of the optical generating pulse rather than a constant dc level. Alternatively, this generation mechanism can be understood by considering the fact that the optical pulses have significant bandwidths. Thus, the high-frequency components can mix with the low-frequency components within a given pulse to produce a pulse at the difference frequency (Fig. 4.2). Since the optical pulses have a bandwidth of a few THz, the difference frequencies fall in the THz range. Difference frequency mixing produces a low frequency polarization which follows the envelope of the incident laser pulse.

Fig. 4.2. The low frequency component of a pulse ω1, and the high frequency component

ω2, are mixing, producing a pulse at the difference frequency ω3=ω1-ω2.

One advantage of optical rectification is that it is a nonresonant method and the THz pulse width is limited only by the optical laser pulse width (and the phonon-mode absorbtion of the crystal), and not the response time of the material. Some of the shortest THz pulses to date, with bandwidths up to 100 THz and beyond have been generated in this or a similar fashion [14-16]. Kubler et. al. have used a 20 µm thick GaSe crystal to generate THz pulses with a bandwidth beyond 120 THz, [14]. In a THz spectrometer, the generated radiation bandwidth can be tuned between 2.5 THz and 100 THz, by choice of the appropriate nonlinear medium and optical generating pulse length. Once the THz pulses have been generated, the need for a reliable detection scheme becomes a must. In the THz spectroscopy development period a multitude of detection

113 methods have been constructed. The oldest one is the photoconductive antenna [3-5, 14], based on low-temperature-grown GaAs and Si-GaAs. When a high-power low-repetition rate experiment or a destructive experiment is performed, a single-shot detection is desirable (it is possible to collect an entire THz waveform without having to scan a delay line) [17-20]. Nahata and Heinz have demonstrated that it is possible to detect THz pulses via optical second harmonic generation, [21]. Polarization modulation, as opposed to amplitude modulation, when using optoelectronic detection has also been demonstrated, [22].

Fig. 4.3. Schematics of free space electrooptic sampling detection. PBS – pellicle beam splitter.

In our setup we are detecting THz pulses via free space electrooptic sampling (FSEOS). The reason is that when using an amplified laser system (Ti:Sapphire in our case), the pulses are detected best via free space electrooptic sampling rather than photoconductive dipole antennas (PDA’s) [15, 23]. It has the advantage that it is a nonresonant method of detection, so the potential for damaging the detector crystal with the focused readout beam is much lower. The FSEOS is based on the electric field of a THz pulse inducing a small birefringence in an electro-optic crystal through a non-linearity of the first order (Pockels effect). Passing through such crystal, the initially linearly polarized optical probe beam gains a small elliptical polarization. In the first approximation, this ellipticity is proportional to the electric field applied to the crystal, i.e. to the amplitude of the THz pulse at a given moment. Because the THz field is much longer than the optical probe pulse (several ps versus ~150 fs), the THz electric field can be approximated as a dc bias field. Therefore, varying the delay between the THz and optical probe pulse, the whole time profile of the first can be traced. As FSEOS active medium, a variety of dielectric materials like LiTaO 3 [24, 25] and ZnTe [24] or polymers polarized by externally applied field [25, 26] are employed.

114 In practice, a quarter-wavelength plate is placed behind the electro-optic crystal to make the initially linear polarization of the probe beam (at E THz = 0) circular (Fig. 4.3). A Wollaston prism separates its y ” and z ” components and sends them to a differential detector which is connected to a preamplifier and a lock-in amplifier. With no THz present, the components have equal intensity and the differential signal is zero.

4.1.2. Experimental setup for terahertz time-domain spectroscopy.

Our THz spectrometer is driven by a commercial femtosecond Ti:Sapphire (Ti:Sa) amplifier (Hurricane, Spectra Physics). This system is detailed described in §4.2. In the following we will describe the THz time-domain spectrometer based on optical rectification of the 120 fs pulses with the central wavelength of approximately 800 nm in a ZnTe crystal and detection of the THz pulses in another ZnTe crystal using free-space electrooptic sampling (FSEOS) technique (Fig. 4.4). The emitter crystal is [110]-oriented and 1 mm thick. Detection is made using a [110]-oriented ZnTe crystal of 0.2 mm mounted on a [100]-oriented ZnTe crystal of 0.5 mm. When a short microwave pulse is applied on the detection crystal, the group-velocity mismatching of ZnTe must be considered. The measured group-velocity mismatching of ZnTe is 0.4 ps/mm. In order to reduce walk-off for better temporal resolution, the thinner [110] crystal provides shorter convolution window (about 0.1 ps), and the thicker [100] crystal delays the reflected THz pulse, with no contribution to the electrooptic phase retardation. In brief, it the combination of these two crystals allows to avoid the reflection of the THz pulse during the detection. The disadvantage of using a thinner crystal is the reduction of electrooptic signal, due to the shorter interaction length, [24]. The 150 fs laser pulse with the central wavelength around 800 nm excites a transient nonlinear polarization in the emitter ZnTe crystal, which produces an electromagnetic pulse with the frequency spectrum in the THz range. The THz pulse is guided to the detector ZnTe crystal by the off-axis parabolic mirrors. Once the THz pulse is transmitted to the detection ZnTe crystal, its electric field induces a birefringence in this crystal. This birefringence is detected by the phase retardation of the very weak (less than 3 percent of the Hurricane output power) 800 nm laser probe pulse which has a prealigned polarization. This probe laser pulse is temporally delayed with respect to the pump laser used for THz generation using the variable delay line. It is therefore possible to measure the phase retardation (which is proportional to the electric field of the THz pulse) with a time resolution of about 150 fs. Considering that both THz pump and THz probe laser pulses originate from the same laser beam and normally have the same duration, we can state the

115 fundamental limitation of the THz-TDS method: it is not possible to measure THz signals which are shorter than the probe laser pulse itself. Therefore in a conventional THz-TDS setup the bandwidth of the THz pulse cannot exceed that of the detection probe pulse. After the propagation through the quarterwave plate, the phase-retarded optical probe pulse will become elliptically polarized, which will result in different light intensities incident on the two photodiodes of the differential detector. The difference in voltages detected by these photodiodes will be proportional to the induced phase retardation, and therefore to the electric field strength in the THz pulse. Varying the delay between the THz pulse and the optical probe one can therefore temporally sample the electric field in the THz pulse.

Fig. 4.4. Schematic representation of the terahertz time-domain transmission spectrometer.

illuminated with the probe beam, when no THz field is applied. Then ∆V=V max −V max = 0 . n0 and r41 are the refractive index and the electrooptic coefficient of the nonlinear crystal at the frequency of the probe beam ω. d is the thickness of the crystal. The electrooptic signal ∆V in our experiment was first preamplified and then read by a commercial lock-in amplifier SR 830 DSP at the frequency 166.6 Hz corresponding to 1/6 of the Ti:Sapphire amplifier repetition rate. The THz generation beam was modulated at this frequency with an optical chopper. For the ZnTe crystal probed at 800 nm n0=3.22, [27].

116 0.10

0.05

0.00

-0.05

-0.10 electric field (arb. units)

-10 -5 0 5 10 15 20 time (ps)

Fig. 4.5. Time trace of a typical terahertz pulse obtained with our THz spectrometer.

1 amplitude (arb. units) (arb. amplitude

0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 frequency (THz)

Fig. 4.6. Amplitude and phase of the terahertz time trace Fourier transform.

THz generation by optical rectification of ultrashort laser pulses was first demonstrated by the group of Y. R. Shen in 1971 [28]. The FSEOS detection scheme was first introduced by the groups of X. C. Zhang [24] and P. Uhd Jepsen and H. Helm [29] in 1996. Both these techniques are widely used ever since and allow for coherent detection of the electric field

117 temporal evolution in ultrashort transients. Recently the generation and detection of THz pulses with the peak electric field strength of the order of 1 MV/cm was demonstrated using optical rectification and FSEOS detection schemes, [15]. A typical time trace of a THz pulse generated and detected in our THz spectrometer is shown in Fig. 4.5, while it’s amplitude and phase frequency dependent spectra are shown in Fig. 4.6. This pulse was generated by the 800 nm central wavelength , 150 fs long laser pulse with the fluence of approximately 0.5 mJ/cm 2. The THz pulse has a relatively complicated shape with a main single-cycle oscillation, followed by decaying anharmonic oscillations. The maximum signal-to-noise (S/N) ratio of this pulse is about 10 3. This THz pulse has a useful bandwidth in the range 0.2-2.5 THz, which is considerably smaller than the bandwidth of the 150 fs excitation laser pulse. The temporal shape and bandwidth of a THz pulse produced by optical rectification in an optically transparent nonlinear crystal is determined by the temporal shape and bandwidth of the excitation pulse as well as by the phase mismatch between the optical and THz pulses copropagating through the crystal and the THz absorbtion by phonons in the crystal.

4.1.3. Propagation of an electromagnetic wave packet through the medium. Terahertz spectral analysis.

In the following we will concentrate on the propagation of a linearly polarized electromagnetic signal through a plan-parallel slab of dispersive medium, i.e. a medium with frequency-dependent refractive index and absorbtion coefficient. Fig. 4.7 gives a schematic representation of the propagation of a THz pulse through the sample with frequency dependent refractive index n( ω) and power absorbtion coefficient α(ω). The free- space signal Ê r only suffers changes caused by the optical components encountered during propagation. We call this change the system response. The sample signal Ê s, however, does experience, besides the system response, reflection losses at the sample’s interfaces as well as absorbtion and chirp inside the sample. The extraction of the dielectric properties of the sample, such as refractive index and power absorbtion coefficient requires a change from the time-domain to the frequency-domain, which is done using Fourier transformation. In the following equations all parameters are frequency-dependent. The detected THz signals propagated through the free-space Ê r and through both free-space and sample Ê s have a complex form:

118

Fig. 4.7. Ê r and Ê s are the THz signals transmitted through the free space (reference pulse) and through the free space and sample. R is the length of the optical path between emitter

and detector and d is the sample thickness. r 01 , t 01 and r 10 , t 10 are the amplitude reflection and transmission coefficients at the front and the back of the sample, respectively. n( ω) and α(ω) are the frequency dependent refractive index and power absorption coefficient of the sample.

) ikr iφ0 Er = E0e = E0e ) ( ik (r −d ) ni kd iφsample Es = E0t01 t10 e e = Ese , (4.1)

In this equations E0 is the electric field strength of the reference pulse when detected, k= ω/c is the space wavevector when there is no sample, r is the length of the optical path, d is the sample thickness, ň=n+i κ is the sample complex refractive index, and t01 and t10 are the amplitude transmission coefficients at the front and back surfaces of the sample,

119 respectively. The electric fields Ê s and Ê r in complex form are defined by their amplitude and phases E s, φsample and E 0, φ0. The amplitude reflection and transmission coefficients at the front and back surfaces of the sample in the case of normal incidence are: ( 1− n r01 = ( t01 =1+ r01 1+ n ( n −1 r10 = ( t10 = 1+ r10 , (4.2) 1+ n

Knowing that the power absorbtion coefficient α= 2k κ and taking into account Eq. 4.2 we can write the ratio of the electric fields Ê s and Ê r as:

( α Eˆ 4n − d ( E i(φ −φ ) s = e 2 ei(n − )1 kd = s e sample 0 , (4.3) ˆ ( 2 Er 1( + n) E0

Performing a complex fit of the experimentally obtained transmission with the theoretical one displayed in Eq. 4.3, the real and imaginary refractive index, n and κ are obtained. The real refractive index and the absorbtion coefficient, α, can also be obtained considering the following approximation: neglecting the losses suffered due to absorbtion at the air - sample interfaces (i.e. taking κ=0 in the first term of Eq. 4.3). This approximation is valid in the case of relatively low-absorbing samples of any thickness, or optically thick but relatively high-absorbing samples. In this case, the refractive index in Eq. 4.3 will become real (the absorptive component κ is neglected) and the transmission becomes:

α Eˆ 4n − d E i(φ −φ ) s = e 2 ei(n− )1 kd = s e sample 0 , (4.4) ˆ 2 Er 1( + n) E0

Therefore from the measured phase difference of the sample and reference pulses, and taking into account that ω= 2 πf, f being the frequency, one can obtain the frequency dependent real part of the refractive index: φ − φ n = sample 0 + ,1 (4.5) 2πf d c

120

Having calculated the refractive index one can separate the reflection and absorption losses in the sample. The power absorption coefficient of the sample will be then:

2  E 1( + n)2  α = − ln  s , (4.6) d  E0 4n 

All the results presented in this work have been obtained using a procedure which does not employ the last mentioned approximation (neglecting the absorption losses at the sample - air interfaces). We have chosen to perform a fit to the experimentally obtained transmission with the theoretical transmission.

In the following we present the results of THz-TDS on a 0,73 mm thick Cu 2O crystal at room temperature, using a 1 mm thick [110]-oriented ZnTe crystals for emission and a double ZnTe crystal: a 0.2 mm [110] oriented crystal glued on a 0.5 mm [100] for detection. Fig. 4.8 shows the reference and the sample THz transients obtained at room temperature. The sample THz pulse’s zero level was manually displaced from that of the reference in the figure for clarity. The time t=0 has been chosen to be the time of the maximum amplitude of the main peak. One can notice that the main feature of the sample signal appears approximately 4 ps later than the main feature of the reference pulse, so one could immediately estimate nCu 2O(THz) using the simple relation ∆t=(n − 1)d/c (where d is the sample thickness and c is the speed of light in vacuum) to find n=2,7. This sample pulse delay results from the propagation through the sample, who’s refractive index is greater than 1. It has smaller amplitude and its shape is distorted in comparison to the reference pulse. This results from the frequency dependent absorption coefficient and refractive index of the Cu 2O sample crystal. In the sample signal, the main THz peak at 4 ps is followed by another feature at 10 ps distance. This feature originates from the etalon reflection in the plan parallel sample, i.e. it is a part of the main THz signal after it made a round trip in the sample. Knowing the sample refractive index we have calculated that in order to travel two sample lengths, the THz reflected pulse will need approximately 9,82 ps. The etalon reflection might be suppressed or removed by using a wedged sample. This echo signal has a distorted time shape and is much weaker than the main pulse. Nevertheless, if Fourier transform of the THz transient and the echo is performed, the echo will influence the transform, leading to an incorrect spectrum. In order to avoid this inconvenience, one can

121 zero the echo part of the time-domain THz signal, since the oscillations from the main pulse have already disappeared in the noise at this time. The time-domain THz signal with a zeroed echo part is shown by the solid line in Fig. 4.8. Fig. 4.9 presents the amplitude spectra of the sample THz pulses. The amplitude spectra have their maxima at around 1 THz. The spectra have a cut-off frequency of approximately at 0,2 and 2,3 THz. Electric field Electric

-2 0 2 4 6 8 10 Time, ps

Fig. 4.8. THz-TDS on a 0,73 mm thick Cu 2O ([100] orientation) crystal at room temperature.

Amplitude

0.0 0.5 1.0 1.5 2.0 2.5 Frequency, THz

Fig. 4.9. Frequency domain amplitude of THz pulse transmitted through the 0,73 mm thick

Cu 2O crystal of [100] orientation.

122 Fig. 4.10 and Fig. 4.11 shows the refractive index and absorbtion spectra of the Cu 2O crystal at room temperature. These refractive index and absorption spectra are calculated from the amplitude and phase spectra using the formulas Eq. 4.6 and Eq. 4.7. The spectra show useful information in the range 0,2-2,3 THz. A definition of the spectral limits of reliability for the obtained information can be done taking into account the frequency- dependent dynamic range (DR) of the amplitude spectra. As was shown by Jepsen and Fischer [30], one can define the frequency-dependent dynamic range of the measured absorbtion spectrum based on the signal-to-noise consideration. The maximal frequency- dependent absorbtion coefficient that can be reliably measured with THz-TDS follows from Eq. 4.7 and is calculated using the formula:

2  4n  αmax = − ln DR , (4.8) d  1( + n)2 

Fig. 4.10. Frequency dependent refractive index of the Cu 2O crystal of [100] orientation at 17 K calculated from the amplitude and phase spectra.

123 4 ) -1 cm (

2 Absorbtion coefficient Absorbtion 0

0.5 1.0 1.5 2.0 Frequency (THz)

Fig. 4.11. Power absoprtion coefficient of the Cu 2O crystal of [100] orientation at 17 K calculated from the amplitude and phase spectra. where d is the sample thickness, DR is the frequency-dependent dynamic range, and n is the frequency-dependent refractive index of ~2,3. The frequency resolution of the THz-TDS is given by the time step and the number of data points in the time-domain measurement ∆f =1/Ndt , where N is the number of data points in the THz time-domain signal and dt is the time step. The fast Fourier transform method (FFT) is used to perform the Fourier transforms numerically. Its performance is best when the number of data points N is a power of 2. In the measurements presented above the raw data consists of 625 data points with a time step of 40 fs, thus making the whole time scan of 25 ps. In order to reach the nearest power of 2 value which is 1024, the time-domain signals were padded with zeroes after the last measured data point. The reference signal was padded with zeroes already after the 532 - th data point in order to avoid the Fabry-Perot effects in the frequency domain caused by the echo of the sample (Fig. 4.8). This operation did not affect the integrity of the measured and calculated data, because in this range there was already no signal above the noise anymore. Thus, the frequency resolution of our experiment was ∆f =0,04 THz. It should be mentioned that the zero padding and removal of the echoes in the time-domain data should be performed very carefully because if the meaningful part of the original signal is cut, the frequency domain data will contain false features and lack the true ones.

124 4.1.4. Time-resolved optical-pump terahertz-probe spectroscopy.

A time-resolved optical-pump THz-probe experimental setup is shown in Fig. 4.12. An intense optical pump pulse, which is derived from the same laser beam that triggers the THz transmitter and detector, photoexcites the sample. Using a mechanical delay line, the optically induced changes in the transmitted electric field can be measured with subpicosecond resolution. The complex sample signal as a function of frequency and pump- probe delay time τ is given by:

FFT (E (t) + ∆E t,( τ )) E (ω,τ ) S(ω,τ ) = eq = sig , (4.9) FFT (Eref (t)) Eref (ω)

where Eeq (t) is the equilibrium scan of the sample in the time-domain without optical excitation, ∆E(t, τ) = E ex (t, τ )-Eeq (t) is the induced change in the electric field with Eeq (t, τ) the scan of the sample with optical excitation.

Experimentally, ∆E(t, τ= τ ι) is obtained by scanning the THz probe delay line and mechanically chopping the optical pump delay line which is positioned at a specific pump- probe delay time τ= τ ι. This procedure measures the difference between Eex (t, τ= τ ι) and

Eeq (t) at the rate of the chopper frequency immediately yielding ∆E(t r,τ=τ ι) at each THz probe delay time tr. Alternatively, by chopping and scanning the THz probe delay line one would collect the data for Eex (t, τ= τ ι) (i.e. pump on) and Eeq (t) (i.e. pump off) in separate scans. The direct measurement of ∆E(t, τ) allows for increased signal sensitivity as it is more robust to system drift, particularly when the induced change is small. Then the real and imaginary part of the dielectric function can be obtained through a procedure which was given in the previous section. After obtaining ∆E(t, τ= τ ι) , Eeq (t) and Eref (t) are measured by chopping and scanning the THz probe delay line. ∆E(t, τ= τ ι) must be measured at each pump-probe delay time because the optical excitation can induce changes in both the phase and amplitude of the THz field. However, for samples in which the optical excitation produces changes primarily in the amplitude of the THz electric field, an alternative method can be used. ∆E(t peak ,τ) as a function of τ is obtained in a single scan by chopping and scanning the optical pump delay line while the THz delay line is positioned at the peak of the THz electric field. This method, if applicable, can provide considerable time savings in measuring the induced dynamics. It must be kept in mind that this proceeding is

125 an approximation which can be employed only when the induced change is small, or when the phase of the THz pulse does not change with optical excitation. This method is in most of the cases not applicable, because t peak is usually an unstable point. Time τ=0 is chosen as the pump pulse delay where an induced change in the THz pulse amplitude is first observed. The THz electric field is four orders of magnitude lower than the excitation pulse electric field. Therefore, one can be sure that the THz pulse acts as a true probe pulse, not perturbing the system. One of the advantages of the THz-TDS is the distinctive detection method [31, 32]: both, the change in absorbance and the phase shift of all frequency components contained in the probe field are determined. This information can also be expressed in terms of complex conductivity spectrum σ(ω,τ) = σ’( ω,τ) + i σ’’( ω,τ), which is a function of the time τ after photoexcitation. The experimentally determined absorbtion and phase shift of the THz electric field ∆E(t, τ) fully determine σ(ω,τ), [33]. However, when σ(ω,τ) changes on a time scale τ which is comparable or shorter than the THz pulse duration, the extraction of σ(ω,τ) from ∆E(t, τ) becomes more problematic, because different sample properties are probed during the time span of the THz pulse, [23, 33]. This effect must be taken into account in the data analysis.

Fig. 4.12. Typical optical setup in an optical-pump THz-probe experiment. An optical pump pulse directly excites the sample, and the focused THz beam probes the far-infrared properties of the sample as a function of time after optical excitation.

126 An optical-pump THz-probe experiment on Cu 2O reveals that the optical excitation induces changes in both, the phase and amplitude of the THz field. In this situation, as discussed in the beginning of this section, (t, τ) must be measured at each pump-probe delay. Fig. 4.13 presents a schematic representation of the two dimensional analysis performed on the measured Cu 2O data. In the bottom part is the THz pulse before the arrival of the excitation pulse. The top plot represents the the THz pulses E(t, τ) at different times τ after the excitation. A decrease in the amplitude of E(t, τ) with τ is observed, indicating a decrease in conductivity with the pump-probe delay. There is also a shift of the 0 waveform to the left. The dashed line at a 45 angle represents the path of the pump pulse in the measurement: each point in a horizontal cross section E(t) has a different pump-probe delay. Transforming the data along this line introduces an alternative time τ’ to describe the delay between the excitation pulse and all points on the probe THz pulse with the same pump delay. An alternative to the two dimensional data analyze is to experimentally make sure that the delay between the optical excitation pulse and the THz probe pulse remains constant during the THz pulse measurement. This would significantly shorten the data analysis process, as well as the time duration of the measurement itself. In a classical optical-pump, terahertz- probe experimental setup, two delay stages are used. One of them, usually placed on the terahertz probe beam is varying the delay between the terahertz probe beam and the terahertz generation pulse (time t in our notation, see Fig. 4.14). The second delay, usually placed on the optical excitation pulse, is varying the time between the terahertz generation pulse and the optical excitation pulse (time τ in our notation). There is a possibility to perform the THz measurements keeping the delay between the optical-pump pulse and the terahertz probe pulse, τ’, constant, while moving them both relative to the terahertz generation pulse (in Fig. 4.14, this movement is represented by a dashed line). This possibility does not require any change in the classical setup. Without changing the setup, one can simply move simultaneously both delay stages with the same speed during the THz pulse measurement. This will ensure a constant τ’.

127

Fig. 4.13. A measurement taken on a copper oxide sample of [100] orientation. In the bottom part is the THz pulse before the arrival of the excitation pulse. The top plot represents the the THz pulses E(t, τ) at different times τ after the excitation. A decrease in the amplitude of E(t, τ) with τ is observed, indicating a decrease in conductivity with the 0 pump-probe delay. There is also a shift of the waveform to the left. The dashed line at a 45 angle represents the path of the pump pulse in the measurement: each point in a horizontal cross section E(t) has a different pump-probe delay. Transforming the data along this line introduces an alternative time τ’ to describe the delay between the excitation pulse and all points on the probe THz pulse with the same pump delay.

128

Fig. 4.14. Pulses which are used during an optical-pump terahertz-probe experiment and the time delays between them. The time delays have the same notations as in the 2D data processing section.

4.2. Induced terahertz response in Cu 2O.

A Cu 2O sample of [100] crystallographic orientation, 8 mm in diameter and 0,73 mm thickness, has been used to perform terahertz time-domain (THz-TDS) and optical-pump terahertz-probe transmission measurements. The sample was kindly provided by A. Revcolevschi (University of Paris IV). The setup used for this experiment as well as a detailed measurement description are presented in §6.1. An 800 nm infrared beam (1,2 2 mJ/cm ) induces a change in the THz pulse transmission through the sample. The strongest induced absorbtion takes place at low temperature, and is depicted in Fig. 4.15. This measurement monitors the transmitted THz pulse peak amplitude, while changing the delay between the optical excitation pulse and the THz pulse. We have chosen τ=0, the time where a change in transmission is first observed. The transient THz response shows three regimes in time: 1) a sharp decrease immediately after τ=0 (<8 ps).

2) a fast initial recovery with time constant τ1=13 ps.

3) a slow recovery with time constant of τ1=425 ps.

129 Fig. 4.16 displays typical dielectric spectra in each of these time regimes. The top two panels show ε1(ω) and ε2(ω), respectively, for t=6 ps, showing a clear broad absorption feature in ε2 centered around 0,9 THz, with an accompanying dispersive feature around the same frequency in ε1. During the initial fast recovery, this feature remains present, though its central frequency shifts gradually to lower values (middle panels in Fig. 4.16, t=11 ps). Finally, in the slow recovery regime the peaked absorption feature has disappeared, leaving only rather featureless ε1 and ε2 spectra (lower panls in Fig. 4.16, t=840 ps).

Fig. 4.15. Induced change in THz transmission through Cu 2O of [100] orientation at 18 K, when the sample is excited with an excitation pulse with energy of 1,2 mJ/cm 2.

130

Fig. 4.16. Dielectric function real and imaginary spectra, measured at different times after the excitation moment. The first set of spectra is taken during the fast rise in absorbtion, after 6 ps from the excitation; the second one is taken during the fast exponential decrease in absorbtion, after 11 ps from the excitation; the third set is taken during the slow exponential decrease in absorbtion, after 840 ps from the excitation moment. After 6 and 11

ps an oscillator signature is observed in both ε1 and ε2, indicated by arrows. After 840 ps the oscillator signature is not present anymore.

It is clear at this point that optically exciting the sample with 800 nm infrared pump 2 pulses, having 1,2 mJ/cm energy, at 18 K, results in considerable changes in the sample’s 2 response. In the following we will consider only the excitation with 1,2 mJ/cm energy per pulse. Furthermore, we will concentrate on the first two parts of the change in transmission spectra, namely the sharp rise in absorption and the fast exponential decay which follows immediately after (Fig. 6.15) because we have seen earlier that the most spectacular changes in the spectra occur in these two regions. A few of the measured THz pulses are presented in Fig. 4.17. The pulses are displayed in two plots: the left side presents pulses taken during the sharp rise in absorption; the right

131

Fig. 4.17. Terahertz time traces measured at different time intervals from the excitation time τ=0. The measurement has been performed at 18 K with an excitation pulse energy of 1,2 mJ/cm 2.

Fig. 4.18. Fourier transforms of the terahertz time traces measured at different time intervals from the excitation time τ=0. The measurements have been performed at 18 K with an excitation pulse energy of 1,2 mJ/cm 2.

132 side shows pulses taken during the fast exponential decay in absorption. Considerable changes with time are easily observed in these pulses. Looking at the left side plot, a change with time is noticed in both, the THz pulse amplitude and shape. Second, there is a phase shift towards earlier times of the positive THz peak which is at early times at 0 ps on the time axis, and then shifts to 0,24 ps for the pulse recorded at 5 ps after excitation. A third THz peak, a positive one, starts to develop, positioned around 0,77 ps (most easily seen in the pulse taken at t=5 ps after excitation). Furthermore, a strong decrease in the THz pulse amplitude with time can be noticed. The pulses taken during the fast exponential decay in absorption (right plot in Fig.4.17), show a typical recovery: a shift toward later times of the positive THz peak, and an increase in the THz pulse amplitude with time. By these, the dynamic, presented in Fig. 4.16, is partially due to the shift and change in THz pulse amplitude at the same time. The Fourier transformations of these pulses are presented in Fig. 4.18. An absorption peak is observed first at 3 ps after the photoexcitation, and it is shifting towards lower frequencies until around 8 ps after excitation. A two dimensional terahertz spectral analysis has been performed on the time traces as described in §6.1. As a result two main quantities have been extracted: not only the change in absorbance, but also the phase shift of all frequency components contained in the probe field. This information can be expressed in terms of complex dielectric function spectrum

ε(ω,τ)= ε1(ω,τ)+i ε2(ω,τ), or in terms of complex optical conductivity spectrum

σ(ω,τ)= σ1(ω,τ)+i σ2(ω,τ), which depend parametrically on the time τ after photoexcitation. These quantities are depicted together in Fig. 4.19.

We will focus in the following on the induced changes in ε2(ω,τ) which is directly related to the absorption. Fig. 4.19 displays in the upper right part ε2(ω,τ) at different times (between 3 and 16 ps) after photoexcitation. The open circles represent the sample terahertz response before photoexcitation. Two striking effects can be immediately observed. First, a broad mode is nearly instantaneously present in the ε2 spectra, centered around 1,3 THz, which gradually sharpens, weakens and shifts to lower frequency upon increasing time after photoexcitation. Eventually, this “node” disappears around 16 ps. Second, a broad feature appears with increasing strength toward lower frequency, which spectral weight gradually shifts toward lower frequency upon increasing time after photoexcitation. This feature persists even after 16 ps, and is reminiscent of a Drude-like free electron response, as has also been observed in other semiconductors.

133

Fig. 4.19. The real and imaginary parts of the dielectric function and optical conductivity, measured at different times after photoexcitation. The measurements have been performed at 18 K with an excitation pulse energy of 1,2 mJ/cm 2. The open circles represent the sample response without optical excitation.

4.2.1. Power dependence.

2 We have shown that exciting the Cu 2O sample with 800 nm pulses of 1,2 mJ/cm energy, induce a change in absorbance which can be intuitively be explained as the effect of two contribution: a Drude like response attributed to free carriers and a second contribution which appears as an oscillator response with not clear origin at this point. To gain more insight into the oscillator response, we have measured the THz spectra at 6,5 ps after photoexcitation as a function of pump power. Fig. 4.20 displays the transmitted terahertz transients power dependence measured at 6,5 ps after photoexcitation. This particular time

134

Fig. 4.20. Power dependence of the transmitted terahertz pulses, measured at 18 K. The measurements have been performed at 18 K and at τ=6.5 ps after photoexcitation.

Fig. 4.21. Power dependence of the real part of the optical conductivity, σ1. The measurements have been performed at 18 K and at τ=6,5 ps after photoexcitation.

135 has been chosen because here the Lorentz shaped response is fully developed and the Drude + oscillator behavior is easily recognizable. The results are presented, for convenience, in terms of the real optical conductivity σ1 which is directly related to the imaginary part of the dielectric function. The transients show a gradually increasing change upon increasing pump power, until the changes saturate around 0,8 mJ/cm 2. Fig.4.21 shows the power dependence of the real part of the optical conductivity. Clearly, the oscillator like response shifts toward lower energy and loses spectral weight upon decreasing pump power, until it finally vanishes for powers below 0,8 mJ/cm 2. The observed peak in the real part of the optical conductivity can be characterized by its spectral weight, its peak position and its spectral width. The power dependence of these parameters are displayed in Fig. 4.23, 4.24 and 4.25, respectively. The integrated area (Fig. 4.22) and the peak position (Fig. 4.23) increase linearly with the excitation pulse energy. The width does not vary a lot, remaining approximately constant around 0,6 THz (Fig. 4.24).

Fig. 4.22. Integrated area below the peak, evolution with excitation pulse energy. Measurements have been performed at 18 K and at τ=6,5 ps after photoexcitation.

136

Fig. 4.23. Peak position evolution with the excitation pulse energy. Measurements have been performed at 18 K and at τ=6,5 ps after photoexcitation.

Fig. 4.24. The peak width evolution with excitation pulse energy. Measurements have been performed at 18 K and at τ= 6.5 ps after photoexcitation.

137

4.3. Preliminary discussion.

4.3.1. Possible theoretical considerations.

We noticed a change in Cu 2O sample absorption at 18 K, upon optical excitation with 800 nm light. This excitation triggered two phenomena: the creation of some particles by a two-photon excitation process (since the energy gap is 2,17 eV) and a local excitation process (creation of another type of particle) which is viewed as a resonance absorption peak in the far-infrared region. The Drude like response can be well explained by the creation of the free carriers in the system. The main difficulty is to understand the origin of the origin of the induced absorption peak. In general, this narrow absorption line can be described as an oscillator response. The intensity of the absorption peak is decreasing with time (in a few picoseconds) and shifts towards lower energies: from 3 to 1,5 meV. The 2 power dependence of the absorption peak strength is linear up to 0,6 mJ/cm with a saturation above this power density value. An interesting observation is that the same shift of the oscillator towards lower energies with decreasing the pump power density is present. Another unexpected observation, although one would like to see this type of measurement reproduced, is a step like behavior with the increase in temperature: while below 100 K, the absorbtion peak is clearly visible, somewhere between 100 K and 150 K it completely disappears. Our measurements have been performed on two different Cu 2O samples. The difference of this samples lies in the amount of oxygen and copper impurities, which relative concentration can be easily determined from the luminescence experiment (§1.3). As one may see from Fig. 1.6, there is a luminescence band centered around 820 nm, attributed to the charged oxygen vacancy. This charged vacancy state can be formed by taking out the neutral oxygen ion from the lattice. Owing to its double effective positive charge, it may bind one electron resulting in a V O1+ . Optical pump terahertz-probe experiments have shown different sample dielectric responses in the terahertz range: 0,2 to 2,5 THz for different samples: the induced absorption seems to live longer in the sample with high oxygen vacancy concentration. The both last results, even with the linear power dependence, seem to indicate that the observed induced absorption peak might be impurity

VO1+ related. One question might naturally follow: what physical process can be responsible for the observed resonance? In direct-gap semiconductors, the sample response might change dramatically, when the photo-excited carrier spatial distribution is inhomogeneous (Fig. 4.25a). For example, when the photoexcited carriers density is high, they might condensate

138 and form electron-hole liquid droplets, [2]. The droplet phase response of the direct-gap semiconductor might then reveal a resonance in the dielectric function. It has been shown that in Ge there is a resonance feature of the effective dielectric function εeff at the frequency 2 1/2 ωp/√3, [41]. Where ωp=(4 πne /εbm) is the plasma frequency, and: n, m and e are the effective carriers density, mass and electric charge, εb is the sample dielectric function before photoexcitation. Let’s try to apply these considerations now to Cu 2O . For an electron-hole droplet, the plasma frequency can be scaled with the exciton binding energy 3 1/2 through ħωp=E ex (12/r s ) , where Eex is the exciton binding energy and rs is a dimensionless 3 1/3 parameter defined as rs=(3/4 πna B ) , [2]. Here n is the exciton density and aB the Bohr -10 radius. For Cu 2O , aB=7·10 m and the yellow exciton binding energy is about 150 meV while the green excitonic binding energy is about 90 meV. Simple calculations show that in order to reach the electron-hole droplet phase in Cu 2O one would need to create an exciton 21 -3 density of at least 10 cm . During our experiments we have been able to obtain a density 15 -3 15 -3 of only 10 cm . For an electron density of 10 cm , using the last two formula’s we estimate a plasma frequency of about 500 meV, which is far away from our THz spectrometer range. Starting from the exciton binding energies, in order to obtain a plasma frequency, in our range, of 3 meV (in the case of yellow exciton) or 2 meV (in the case of green exciton) necessary for an electron-hole droplet phase, we need an exciton density of 36 -3 10 cm . These high excitonic densities are impossible to reach using our set-up.

Therefore, we conclude that the resonance observed in Cu 2O after photoexcitation cannot be the response of an electron-hole droplet phase.

139

Fig. 4.25. Discussion about possible processes, which might give an induced response in the far-infrared region: a) electron-hole droplet formation; b) ecitons for the excitonic molecule (biexciton); c) bound biexciton to the cristal imperfection; d) optically induced dipoles (electrons plus charged oxygen vacancy center).

Considering the energy scale at which the resonance appears, namely a few meV, another possibility captured our attention. Looking for particles which have their binding energy comparable with meV, we found out that, in Cu 2O , the possibility exists to create an excitonic molecule (Fig. 4.25b). This would be composed from two excitons and their binding energy would be around 3,3 meV for the biexciton formed by two yellow excitons, and around 16 meV for a biexciton formed by two green excitons, [42]. These energies fit well with the energy where the resonance is observed in our experiment. Exciting with 1,55 eV pump pulse energy we can create excitons only via two-photon absorption to the blue excitonic series with a consequent relaxation into the yellow and green one. Therefore, one might expect quadratic behavior of the absorption peak strength with respect to the power

140 density, which is, as a matter of fact linear (Fig. 4.22). Moreover, if the resonance would be the response of an excitonic molecule gas, then a resonant excitation of the excitonic series (yellow, green, blue and indigo) would result in an enhancement of the exciton density and a stronger dielectric response. Surprisingly, not only that the dielectric sample response was not enhanced upon resonant excitation, but the resonance was not observed anymore. However, the results are not reliable, since the power density in experiments with different pulse energies was different. From another point of view, since the sample dielectric response can be impurity related, another possibility occurs: a dense bi-excitonic gas bound around the impurity centers V O+ (Fig. 4.25c). This idea is consistent with the observed type of power dependence: a linear power dependence at low excitation pump pulse followed by a saturation at high excitation pump pulse (Fig. 4.22). Based on the experiments discussed above, using resonant excitation to the yellow and green excitonic series, we can preliminary conclude that the observed resonance cannot be exciton related. Unfortunately, the clean experimental prove needs to be done to state something definite. Since the induced response is present when the excitation is 1,55 eV, we might be directly exciting a weakly bound impurity state, which contains a resonant transition in the THz range. For example, using the incident photon of approximately 1,55 eV energy, one may excite an electron to the V O+ impurity state, and create a dipole in the system (Fig. 4.25d). Spatially distributed dipoles will interact through the dipole-dipole interaction. For preliminary calculation of the interaction energy, we can assume the dipole moment to be on the order of the unit cell size. For an interaction energy being in the discussed range, the 18 -3 concentration of oxygen vacancies, V O+ , should exceed 10 cm . This value is way below 15 -3 than calculated in [43] and is around 10 cm .

4.3.2. Mathematical description: Drude + Lorentz model.

Let’s try now to build a mathematical description able to roughly describe our experimental observations. We have shown that the experimental frequency dependent absorption spectra are composed from two main contributions: free carriers like contribution and some local oscillator component. In the following we will describe the free carrier contribution by a Drude like dielectric response while the local oscillator contribution can be described by a Lorentz oscillator dielectric response.

2 2 A(ω0 − ω ) Fdτ coll ε1(ω) = εb + 2 2 2 2 − 2 2 , (4.10) ()ω0 − ω + ω γ 1+ ω τ coll

141 1.0 6.5ps 0.8 8ps drude drude lorentz 0.6 lorentz d+l fit d+l fit

2 0.5 2 0.4 ε ε

0.2

0.0 0.0 0.5 1.0 1.5 0.5 1.0 1.5 frequency (THz) frequency (THz) 0.8 9.5ps 10ps 0.6 drude drude lorentz 0.6 lorentz d+l fit d+l fit 0.4 2

2 0.4 ε ε

0.2 0.2

0.0 0.0 0.5 1.0 1.5 0.5 1.0 1.5 frequency (THz) frequency (THz)

0.6 11.25ps 0.6 13.25ps drude drude lorentz lorentz

0.4 d+l fit 2 0.4 d+l fit ε 2 ε 0.2 0.2

0.0 0.0 0.5 1.0 1.5 0.5 1.0 1.5 frequency (THz) frequency (THz)

Fig. 4.26. A few examples of fits with the Drude + Lorentz mathematical description. The open hexagons represent experimental data and the black lines a fit to the Drude + Lorentz mathematical description. The open circles stand for the Drude contribution to the dielectric function, while the open stars stand for the Lorentz contribution.

142 where εb is the background dielectric function, A the oscillator strength, ω0 is the eigen 2 frequency, γ is the dumping parameter, Fd= ωp is the plasma frequency squared and τcoll is the collision time of the free electrons. Using the same notations then the absorptive part of the dielectric constant will be:

Aωγ Fdτ ε 2 (ω) = ε b + 2 2 2 2 − 2 2 , (4.11) ()ω0 − ω + ω γ ()1+ ω τ coll ω

We fitted the experimental data using the Drude+Lorentz mathematical description. Fig. 4.26 shows few examples of fits for different time delay after excitation. Moreover, a series of different parameters have been resulting from the fitting. Three of them are characteristic to the Lorentz component of the imaginary dielectric function: the peak position ω0, strength A and width γ. Other two parameters are characteristic to the 2 imaginary dielectric function of Drude component: the plasma frequency squared Fd=ω0 and collision time τcoll . In the following, we present the time evolution of all of them. Fig. 4.27a describes the peak position as a function of time after excitation. The position moves between 2 THz and 0,5 THz while the time evolves between 3 and 13 ps. An exponential decrease manages to decently describe this movement. The solid line represents a data fit to a first order exponential decay with a lifetime τ=3,5 ps. Integrated area below the pulse is directly related to the peak intensity or strength. This area which was calculated and is displayed in Fig. 4.27b, has a behavior similar to the one displayed by the peak position. It exponentially decreases with time τ, after excitation. The data has been fitted again to a first order exponential decay with τ=2,65 ps. From Fig. 4.19 (top right), is visible that the Lorentzian pulse does not only shifts it’s position to low frequencies, but with this shift, his width is also decreasing. For each absorptive part of the dielectric function in Fig. 4.19 (top right), we have approximated the width of it’s Lorentzian component. The time evolution of the peak width is presented in Fig. 6.27c. There is an exponential decrease of the width from 1,2 THz to 0,4 THz. A fit to a first order exponential decay gives us the exponent lifetime τ =3,8 ps. Concerning the “Drude part” of the suggested mathematical description, Fig. 6.27d displays the collision time τcoll taken at different moments after photoexcitation. No significant changes seem to appear in the time interval between 3 and 16 ps after excitation. Fig. 4.27e displays the plasma frequency time evolution after excitation. Like the collision time τcoll , the plasma frequency squared remains constant in the time interval between 3 and 16 ps after photoexcitation.

143 Lorentz parameters Drude parameters 1.5 0.30 2 (a) 1.2 mJ/cm 0.25 18 K (d) 1.0 0.20

(ps) 0.15 τ τ (THz) 0.5 0

0.10 t1=3.54 ω 2 1.2 mJ/cm 0.05 0.0 18 K 0 5 10 15 0 5 10 15 time (ps) time (ps) 30 2 1.2 mJ/cm (b) 25 18 K (e) 20 20

15 d A

10 F 10 t1=1.79 2 5 1.2 mJ/cm 0 18 K 0 0 5 10 15 0 5 10 15 time(ps) time(ps)

1.0 (c)

0.5 (THz) γ

t1=2.19 2 1.2 mJ/cm 0.0 18 K 0 5 10 15 time(ps)

Fig. 4.27 Lorentz and Drude fit parameters versus time τ after photoexcitation: (a) the

peak position, the solid line represents a fit to a first order exponential decay with t 1=3,54 ps. (b) The peak strength, the solid line represents a fit to a first order exponential decay

with t 1=2,65 ps. (c) Peak width, the solid line represents a fit to a first order exponential decay with t 1=3,8 ps. (d) The collision time and (e) plasma frequency time evolution remain constant value.

144 4.4. Conclusion.

In conclusion, we have shown, that the Drude + Lorentz mathematical model describes well the induced terahertz response, observed in Cu2O. Unfortunately, we must say, that at this point there is no straightforward statement, which can be done, in order to explain the observed effects and their time dynamics. Therefore, additional experiments are required to prove or disprove discussed above possibilities, or to propose new ones.

145 References

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147 Conclusions

In this work we presented the study of optical properties of Cu 2O using novel experimental techniques. Using the possibility of the present experimental techniques, we studied the exciton gas in “ultra” regimes: high magnetic fields, low temperatures, high excitation densities. For example, we presented a study of the optical properties of the yellow exciton series in cuprous oxide in high magnetic fields. These experiments provide a unique opportunity to directly determine the thermodynamical properties of the paraexciton gas which could be used as indication of Bose-Einstein condensation of the paraexciton gas. We have demonstrated that an applied magnetic field can be used to reveal the paraexciton emission due to a gentle breaking of the symmetry. For this particular case, the lifetime of excitons is not strongly affected allowing efficient thermalization. From our calculations we determined that the obtained exciton gas density is close to the critical gas density for which the gas is in the quantum regime. The results of the magneto-absorption study suggest that the hydrogen model can be applied only for states with small value of n. Deviations from the simple perturbation theory occur at the higher energies, but already at small magnetic field. For energies higher than the band-gap quasi-Landau levels are observed. This behavior of co-existence of Rydberg and Landau series might be understood in terms of the hydrogen model in high magnetic fields, which for hydrogen would mean fields as high as found in a white dwarf stars. Also, we investigated and discussed the ideas and results of the experiments aimed at determining the exciton gas parameters and their time evolution. It was shown that the lifetime is one of the most critical parameters determining whether a condensate state may occur. For the orthoexcitons, it was shown that the gas adjusts its quantum properties to the density and temperature and that it is in quasi-equilibrium. Apparently the particle decay is faster than the cooling rate, which inhibits a BEC transition. Further, the ways of direct and in-direct probing of the optically inactive paraexcitons were presented and described. Based on the results of the experiments and analysis of the experimental data, we discussed the processes which may contribute to paraexciton losses. The most important conclusion is that at the currently achieved densities (~10 18 cm -3) the main paraexciton loss process is trapping by crystal imperfections. It was concluded, that the lifetime is strongly depends on amount of copper impurities and a mechanism of exciton trapping by copper impurities was proposed. The presented experiments were carried out using quite impure samples: the cooling rate was not sufficient to overcome the particle loss. However, under the proper experimental conditions, it is possible for the paraexcitons to achieve a condensate state: low bath temperature (1,2K for densities ~10 17 cm -3), low concentration of crystal defects,

148 one-photon excitation method for a smaller excitation volume and intense enough excitation source.

The induced response of the Cu 2O in far infra-red energy range can be well-described using the Drude + Lorentz mathematical model. However, for now there is no straightforward statement, which can be done, in order to explain the observed effects and their time dynamics.

149 Samenvatting

In 1931 suggereerde Yakov I. Frenkel dat het mogelijk zou moeten zijn om ladingsdragers te exciteren zonder de elektrische geleiding te veranderen. Deze suggestie kan worden gezien als de geboorte van de exciton en zal later cruciaal blijken in het begrijpen van de optische eigenschappen van vaste stoffen. Een exciton kan worden gezien als de elektron- gat analoog van het waterstof atoom waar het lichte gat de plaats in neemt van het zware proton. Meer dan 20 jaar later waren er twee onafhankelijke experimentele observaties van een waterstof-achtig absorptie spectrum in Cu 2O, in het zichtbare-licht energiegebied door Gross et al. [1], en Hayasi et al. [2]. Deze observatie was het eerste daadwerkelijke experimentele bewijs voor het bestaan van excitonen terwijl Hayasi et al. aanvankelijk de absorptie lijnen niet relateerden aan een exciton. Nu, vele jaren later, is er vele experimentele literatuur te vinden over excitonische eigenschappen van verschillende materialen. Misschien is Cu 2O het best gedocumenteerde geval. Alle groepen van overgangen die theoretisch voorspeld zijn, zijn experimenteel geobserveerd in verschillende gebieden van het meestal zichtbare-licht energiegebied in Cu 2O. Het zal daarom misschien niet verassend zijn dat koper(I)oxide de meest populaire halfgeleider was onder fysici in het pre-silicium era.

Dit werk herbeschouwt de optische exciton eigenschappen van Cu 2O, gebruikmakende van de hedendaagse experimentele mogelijkheden die erg precieze metingen met hoge energie en/of tijdsresolutie omvatten. Hiertoe worden coherente gepulseerde lichtbronnen, zeer sterke magneetvelden en cryogene temperaturen gebruikt. Een kort overzicht van de theorie van excitonen wordt gepresenteerd in hoofdstuk 1. Ook worden er veel-deeltjes effecten zoals Bose-Einstein condensatie behandeld. Algemeen gesproken zijn er een aantal effecten gezien die bewijs leveren voor het feit dat composite-bosons die gemaakt zijn van een even aantal fermionen, zoals He-4-atoms, Cooper paren of alkali-metaal ionen zich onder specifieke condities gedragen als perfecte bosonen en een macroscopische kwantumtoestand kunnen vormen, een Bose-Einstein condensaat. Een exciton, die bestaat uit twee fermionen is een integer-spin deeltje, een composite boson. In dit veel-deeltjessysteem laten deze composite-bosonen allerlei interessante hoge dichtheidseffecten zien, zoals de formatie van excitonische moleculen [3], elektron-gat druppels en plasma formatie en zelfs Bose-Einstein condensatie [4]. Er is een goede indicatie dat Bose-Einstein condensatie van excitonen, dat al werd voorspeld lang geleden [5], daadwerkelijk plaats vindt in dubbel laags tweedimensionale gassen [6]. Hierin worden excitonen gevormd onder evenwichtscondities. Het optreden van Bose-Einstein condensatie van optisch gemaakte excitonen is minder duidelijk aantoonbaar. Alleen in complexe systemen zoals in bijvoorbeeld excitonische polaritonen in halfgeleider

150 microcavities zijn er tekenen te vinden van een gecondenseerde fase. Omdat excitonen in

Cu 2O sterk gebonden zijn behouden ze hun bosonische eigenschappen zelfs bij hoge dichtheden en vanwege de kleine massa van de deeltjes is een condensaat verwacht zelfs bij relatief hoge temperaturen [4]. Het zijn de excitonen die het laagste zijn in energie (singlet ofwel paraexcitonen van de gele exciton groep) waarvan is voorspeld dat ze een condensaat kunnen vormen. Deze excitonen zijn optisch niet actief en hebben daardoor een lange levensduur, mits het kristal van goede kwaliteit is. Een hoge populatie excitonen kan worden bereikt met het laser exciteren van optisch-actieve toestanden die hoger in energie liggen gevolgd door een efficiënte relaxatie naar de laagst gelegen singlet toestand. Hoewel de optische duisterheid van singlet excitons voordelig is aan de ene kant maakt het het wel een stuk moeilijker om ze te meten. Dit is één van de grote obstakels in de zoektocht naar een mogelijke gecondenseerde toestand. In hoofdstuk 2 bediscussiëren we de ideeën en resultaten van de experimenten die als doel hadden de exciton gas parameters en hun tijdsevolutie te bepalen . We laten zien dat de levensduur één van de meest kritische parameters is die bepaald of er een condensaat kan worden gevormd. Er werd ook gezien dat de kwantum eigenschappen van het orthoexciton gas afhangen van de dichtheid en temperatuur van het gas, welke zich in quasi-evenwicht bevindt. Blijkbaar is het deeltjes verval sneller dan de koelsnelheid wat een BEC overgang niet mogelijk maakt. Verder wordt er een manier besproken om de optisch niet-actieve paraexcitonen direct en indirect te meten. Gebaseerd op de experimentele resultaten en analyse bediscussiëren we de processen die bijdragen aan het verslies van paraexcitonen. De meest belangrijke conclusie is dat bij de momenteel behaalde dichtheden (~10 18 cm -3) het meest belangrijke verlies de trapping door kristal imperfecties is. Er werd geconcludeerd dat de levensduur sterk afhangt van de hoeveelheid koper onzuiverheden en om dit te verklaren werd een mechanisme van exciton trapping door koper onzuiverheden voorgesteld. De gepresenteerde experimentele resultaten werden verkregen op samples met relatief hoge onzuiverheid concentraties: de koelsnelheid was niet genoeg om het verlies van deeltjes te compenseren. Echter onder de juiste experimentele condities is het mogelijk voor de paraexcitonen om een condensaat te vormen: lage temperatuur (1,2K voor een dichtheid van ~10 17 cm -3), lage concentratie kristal defecten, één-foton excitatie voor een lagere excitatie volume en een intense excitatie bron. In hoofdstuk 3 presenteren we een studie naar de optische eigenschappen van de gele exciton groep in koper(I)oxide in hoge magneetvelden tot 32T. Wij hebben luminescentie gezien van de optisch niet actieve paraexcitonen als gevolg van de vermenging van de 1s paraexciton toestand met de 1s m J=0 orthoexciton toestand. De experimentele resultaten kunnen goed worden beschreven met eerste orde storingsrekening. We hebben laten zien dat een aangebracht magneet veld kan worden gebruikt om luminescentie te krijgen van

151 paraexcitonen als gevolg van symmetrie verbreking. In dit specifieke geval is de levensduur van de excitons niet erg veranderd door het magneetveld en is efficiënte thermalisatie mogelijk. Afgezien van de directe emissie van de paraexciton toestand, hebben we ook drie andere lijnen gezien welke we hebben toegewezen aan de phonon- geassisteerde emissie. Deze lijnen verschaffen de mogelijkheid om de thermodynamische eigenschappen van het paraexciton gas direct te bepalen. Dit kan worden gebruikt als een indicatie voor Bose- Einstein condensatie van het paraexciton gas. Uit onze berekening hebben we gezien dat de excitongas dichtheid dichtbij de kritische gas dichtheid is waarbij het gas in het kwantum regiem is. De resultaten van de magneto-absorptie studie suggereren dat het waterstof model alleen kan worden toegepast voor toestanden met kleine waarden van n. Afwijkingen van de eenvoudige storingsrekening komen voor bij hogere energieën maar al bij een laag magneetveld. Voor hogere energieën, groter dan de bandgap, worden quasi-Landau niveaus gezien. Het feit dat er op hetzelfde moment Rydberg en Landau levels worden gezien kan worden begrepen in termen van het waterstofmodel in hoge magneetvelden. Dit regiem kan in watersof alleen worden bereikt in zeer hoge magneetvelden en kan worden gezien in witte-dwerg sterren. Tenslotte, het laatste hoofdstuk 4 bediscussieerd verassende observaties in de optische eigenschappen van Cu 2O in de zichtbare-licht-pomppuls, ver-infrarood-meetpuls (ongeveer 1 THz) experimenten. Deze wijken substantieel af van het gewoonlijke Drude gedrag welke wordt gemeten in halfgeleider. The geïnduceerde respons van Cu 2O in the ver-infrarode energie gebied kan goed worden beschreven gebruikmakende van het Drude + Lorentz mathematische model. Echter op dit moment is er geen eenvoudige verklaring voor de geobserveerde effecten en hun tijdsdynamica.

References

1. E.F. Gross and N.A. Karryev, Doklady Akademii Nauk SSSR , 84 , 261 (1952). 2. M. Hayashi, J. Fac. Sci. Hokkaido Univ. , 4, 107 (1952). 3. N. Nagai, R. Shimano, and M.K. Gonokami, Physical Review Letters , 86 , 5795 (2001). 4. S.A. Moskalenko and D.W. Snoke, Bose-Einstein Condensation of excitons and biexcitons . 2000: Cambridge University Press. 5. L.V. Keldysh and A.N. Kozlov, Sov. Physics Solid State , 6, 2219 (1965). 6. J.P. Eisenshtein and A.H. MacDonald, Nature , 432 , 691-694 (2004).

152 Acknowledgements

It happened to be quite a difficult thing for me to write the acknowledgements, in fact, much more difficult than to write any other part of this book. This thesis is the result of four years of my research, but also it is a result of the last four years of my everyday life. I feel thankful to many people for their assistance, contribution and more importantly for their encouragement during these years. I am deeply grateful to all of them and pleased to acknowledge some people who helped to make this goal achievable. I am especially indebted to my mentor Paul van Loosdrecht. I am thankful to him for giving me an outstanding research opportunity consisting of a perfect combination of freedom and directions within this research. Thank you for your support and encouragement that at times has gone beyond the scientific research. I am also grateful for your invaluable patience with me and my manuscript. I am greatly thankful to Audrius Pugzlys, who was not directly connected to my research, but without whom this work would be impossible. Audrius, you always know the right words to inspire me. I wish to express my appreciation to the members of the reading committee, Manfred Fiebich, Makoto Kuwata-Gonokami and Myakzum Salakhov for evaluating and improving this thesis. In the year 2003, in Groningen, I have joined a small (almost a family size) team that has by now grown into a big, friendly and highly professional group: Silviu Sirbu, Dan Cringus, Puri Handayani, Artem Bakulin, Tom Lummen, Michiele Donker, Marian Otter, Pedro Rizo, Daniele Fausti, Filippo Lusitani, Maxim Pshenichnikov, Ben Hesp, Foppe de Haan, Viktor Krasnikov, Dmitry Mazurenko, Ramunas Augulis. I would like to thank Arjen Kamp for the technical support from the beginning. I am indebted to Puri, Marian and Silviu for their valuable contribution to my research. I am grateful to Michiele Donker for his help with this thesis, namely for preparing the “samenvatting” out of a summary. Many thanks to Sonja Groot for help and patience during the unceasing fight against the bureaucracy. I want to express my gratitude to A. Revcolevschi from University of Paris IV and Makoto Gonokami for the samples they provided for my research. Also, many thanks for Nabuko Naka and Mr. Yoshioka for the experiments they carried out in University of Tokyo. I am greatly thankful to Clement Faugeras and Marek Potemski (Grenoble High Magnetic Field Laboratory) for highly fruitful collaboration in experiments on excitons in a strong magnetic field and for cosy atmosphere they provided during my stay in Grenoble.

153 Also, I want to express my appreciation to Sergej Artuhin and Maxim Mostovoy for interests in our results and valuable contribution to the presented work. My deep appreciation goes to David Vainshtein. Thank you for your constant support during these years. My wholehearted gratitude to Denis Markov and Tamara Markova for being... believe me, I smiled when I came to this point in writing the acknowledgements. I know both of you independently and I feel it is not exactly correct to say thanks to you. I would like to say differently: “I’m happy, I met both of you in my life.” I am immensely grateful to Anton Sugonyako, Dan Cringus, Filippo Lusitani, Michiele Donker for open warm hearts and the willingness to help. Without you, my four years in Groningen would be much harder and duller. I also want to express my gratitude to my relatives and friends in Kazan and Moscow. Before I went to the Netherlands, almost 5 years ago, I had not fully understood that distance is indeed a relative thing. You are always in my heart. Dear Yulia, the last period of my research was inspired by you. Forgive me for sharing it with everybody who will probably read these acknowledgements. Your love is a great driving force to me. You are the treasure I was really lucky to find. I am greatly indebted to my parents. Dad, you are the first and permanent mentor in my life. You always taught me to see the beauty of nature, its rules and logic. Mom, your love, trust and constant encouragement – that is what each child can dream of. It is hardly possible to name everyone, who helped me during my research and my time in Groningen, I would like to express my apology and hearty thanks to those, whom I could not mention here.

Dmitry Fishman Groningen, April 2008

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